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",[573,1186,1188,1220],{"className":1187},[576],[573,1189,1191],{"className":1190},[580],[582,1192,1193],{"xmlns":584},[586,1194,1195,1217],{},[589,1196,1197,1199,1201,1203],{},[596,1198,692],{"stretchy":691},[596,1200,697],{},[699,1202,701],{},[1204,1205,1206,1208],"msup",{},[596,1207,704],{"stretchy":691},[589,1209,1210,1212,1215],{},[592,1211,816],{},[596,1213,1214],{},"+",[592,1216,857],{},[602,1218,1219],{"encoding":604},"(-1)^{i+j}",[573,1221,1223],{"className":1222,"ariaHidden":610},[609],[573,1224,1226,1230,1233,1236,1239],{"className":1225},[614],[573,1227],{"className":1228,"style":1229},[618],"height:1.0747em;vertical-align:-0.25em;",[573,1231,692],{"className":1232},[733],[573,1234,697],{"className":1235},[623],[573,1237,701],{"className":1238},[623],[573,1240,1242,1245],{"className":1241},[758],[573,1243,704],{"className":1244},[758],[573,1246,1248],{"className":1247},[880],[573,1249,1251],{"className":1250},[884],[573,1252,1254],{"className":1253},[889],[573,1255,1258],{"className":1256,"style":1257},[893],"height:0.8247em;",[573,1259,1261,1264],{"style":1260},"top:-3.063em;margin-right:0.05em;",[573,1262],{"className":1263,"style":902},[901],[573,1265,1267],{"className":1266},[906,907,908,909],[573,1268,1270,1273,1276],{"className":1269},[623,909],[573,1271,816],{"className":1272},[623,624,909],[573,1274,1214],{"className":1275},[633,909],[573,1277,857],{"className":1278,"style":916},[623,624,909]," nach dem Schachbrettmuster ",[573,1281,1283,1366],{"className":1282},[576],[573,1284,1286],{"className":1285},[580],[582,1287,1288],{"xmlns":584},[586,1289,1290,1363],{},[589,1291,1292],{},[1293,1294,1295],"mstyle",{"scriptlevel":701},[1296,1297,1300,1323,1343],"mtable",{"rowspacing":1298,"columnspacing":1299},"0.1em","0.2778em",[1301,1302,1303,1311,1317],"mtr",{},[1304,1305,1306],"mtd",{},[1293,1307,1308],{"scriptlevel":701,"displaystyle":691},[596,1309,1214],{"lspace":1310,"rspace":1310},"0em",[1304,1312,1313],{},[1293,1314,1315],{"scriptlevel":701,"displaystyle":691},[596,1316,697],{"lspace":1310,"rspace":1310},[1304,1318,1319],{},[1293,1320,1321],{"scriptlevel":701,"displaystyle":691},[596,1322,1214],{"lspace":1310,"rspace":1310},[1301,1324,1325,1331,1337],{},[1304,1326,1327],{},[1293,1328,1329],{"scriptlevel":701,"displaystyle":691},[596,1330,697],{"lspace":1310,"rspace":1310},[1304,1332,1333],{},[1293,1334,1335],{"scriptlevel":701,"displaystyle":691},[596,1336,1214],{"lspace":1310,"rspace":1310},[1304,1338,1339],{},[1293,1340,1341],{"scriptlevel":701,"displaystyle":691},[596,1342,697],{"lspace":1310,"rspace":1310},[1301,1344,1345,1351,1357],{},[1304,1346,1347],{},[1293,1348,1349],{"scriptlevel":701,"displaystyle":691},[596,1350,1214],{"lspace":1310,"rspace":1310},[1304,1352,1353],{},[1293,1354,1355],{"scriptlevel":701,"displaystyle":691},[596,1356,697],{"lspace":1310,"rspace":1310},[1304,1358,1359],{},[1293,1360,1361],{"scriptlevel":701,"displaystyle":691},[596,1362,1214],{"lspace":1310,"rspace":1310},[602,1364,1365],{"encoding":604},"\\begin{smallmatrix} + & - & + \\\\ - & + & - \\\\ + & - & + \\end{smallmatrix}",[573,1367,1369],{"className":1368,"ariaHidden":610},[609],[573,1370,1372,1376],{"className":1371},[614],[573,1373],{"className":1374,"style":1375},[618],"height:1.8em;vertical-align:-0.65em;",[573,1377,1379],{"className":1378},[623],[573,1380,1382,1444,1449,1452,1508,1511,1514],{"className":1381},[1296],[573,1383,1386],{"className":1384},[1385],"col-align-c",[573,1387,1389,1435],{"className":1388},[884,885],[573,1390,1392,1432],{"className":1391},[889],[573,1393,1396,1408,1420],{"className":1394,"style":1395},[893],"height:1.15em;",[573,1397,1399,1402],{"style":1398},"top:-3.43em;",[573,1400],{"className":1401,"style":902},[901],[573,1403,1405],{"className":1404},[623,909,906,907,908],[573,1406,1214],{"className":1407},[623,909],[573,1409,1411,1414],{"style":1410},"top:-2.83em;",[573,1412],{"className":1413,"style":902},[901],[573,1415,1417],{"className":1416},[623,909,906,907,908],[573,1418,697],{"className":1419},[623,909],[573,1421,1423,1426],{"style":1422},"top:-2.23em;",[573,1424],{"className":1425,"style":902},[901],[573,1427,1429],{"className":1428},[623,909,906,907,908],[573,1430,1214],{"className":1431},[623,909],[573,1433,922],{"className":1434},[921],[573,1436,1438],{"className":1437},[889],[573,1439,1442],{"className":1440,"style":1441},[893],"height:0.65em;",[573,1443],{},[573,1445],{"className":1446,"style":1448},[1447],"arraycolsep","width:0.1945em;",[573,1450],{"className":1451,"style":1448},[1447],[573,1453,1455],{"className":1454},[1385],[573,1456,1458,1500],{"className":1457},[884,885],[573,1459,1461,1497],{"className":1460},[889],[573,1462,1464,1475,1486],{"className":1463,"style":1395},[893],[573,1465,1466,1469],{"style":1398},[573,1467],{"className":1468,"style":902},[901],[573,1470,1472],{"className":1471},[623,909,906,907,908],[573,1473,697],{"className":1474},[623,909],[573,1476,1477,1480],{"style":1410},[573,1478],{"className":1479,"style":902},[901],[573,1481,1483],{"className":1482},[623,909,906,907,908],[573,1484,1214],{"className":1485},[623,909],[573,1487,1488,1491],{"style":1422},[573,1489],{"className":1490,"style":902},[901],[573,1492,1494],{"className":1493},[623,909,906,907,908],[573,1495,697],{"className":1496},[623,909],[573,1498,922],{"className":1499},[921],[573,1501,1503],{"className":1502},[889],[573,1504,1506],{"className":1505,"style":1441},[893],[573,1507],{},[573,1509],{"className":1510,"style":1448},[1447],[573,1512],{"className":1513,"style":1448},[1447],[573,1515,1517],{"className":1516},[1385],[573,1518,1520,1562],{"className":1519},[884,885],[573,1521,1523,1559],{"className":1522},[889],[573,1524,1526,1537,1548],{"className":1525,"style":1395},[893],[573,1527,1528,1531],{"style":1398},[573,1529],{"className":1530,"style":902},[901],[573,1532,1534],{"className":1533},[623,909,906,907,908],[573,1535,1214],{"className":1536},[623,909],[573,1538,1539,1542],{"style":1410},[573,1540],{"className":1541,"style":902},[901],[573,1543,1545],{"className":1544},[623,909,906,907,908],[573,1546,697],{"className":1547},[623,909],[573,1549,1550,1553],{"style":1422},[573,1551],{"className":1552,"style":902},[901],[573,1554,1556],{"className":1555},[623,909,906,907,908],[573,1557,1214],{"className":1558},[623,909],[573,1560,922],{"className":1561},[921],[573,1563,1565],{"className":1564},[889],[573,1566,1568],{"className":1567,"style":1441},[893],[573,1569],{},".",[565,1572,1573,1574,1783,1784,1824,1825,1980,1981,1984,1985,2013,2014,2344],{},"Die Summe aller Produkte ",[573,1575,1577,1631],{"className":1576},[576],[573,1578,1580],{"className":1579},[580],[582,1581,1582],{"xmlns":584},[586,1583,1584,1628],{},[589,1585,1586,1588,1590,1592,1604,1614,1616,1618],{},[596,1587,692],{"stretchy":691},[596,1589,697],{},[699,1591,701],{},[1204,1593,1594,1596],{},[596,1595,704],{"stretchy":691},[589,1597,1598,1600,1602],{},[592,1599,816],{},[596,1601,1214],{},[592,1603,857],{},[846,1605,1606,1608],{},[592,1607,850],{},[589,1609,1610,1612],{},[592,1611,816],{},[592,1613,857],{},[592,1615,1103],{},[596,1617,1106],{},[846,1619,1620,1622],{},[592,1621,960],{},[589,1623,1624,1626],{},[592,1625,816],{},[592,1627,857],{},[602,1629,1630],{"encoding":604},"(-1)^{i+j} a_{ij} \\det A_{ij}",[573,1632,1634],{"className":1633,"ariaHidden":610},[609],[573,1635,1637,1641,1644,1647,1650,1688,1731,1734,1737,1740],{"className":1636},[614],[573,1638],{"className":1639,"style":1640},[618],"height:1.1108em;vertical-align:-0.2861em;",[573,1642,692],{"className":1643},[733],[573,1645,697],{"className":1646},[623],[573,1648,701],{"className":1649},[623],[573,1651,1653,1656],{"className":1652},[758],[573,1654,704],{"className":1655},[758],[573,1657,1659],{"className":1658},[880],[573,1660,1662],{"className":1661},[884],[573,1663,1665],{"className":1664},[889],[573,1666,1668],{"className":1667,"style":1257},[893],[573,1669,1670,1673],{"style":1260},[573,1671],{"className":1672,"style":902},[901],[573,1674,1676],{"className":1675},[906,907,908,909],[573,1677,1679,1682,1685],{"className":1678},[623,909],[573,1680,816],{"className":1681},[623,624,909],[573,1683,1214],{"className":1684},[633,909],[573,1686,857],{"className":1687,"style":916},[623,624,909],[573,1689,1691,1694],{"className":1690},[623],[573,1692,850],{"className":1693},[623,624],[573,1695,1697],{"className":1696},[880],[573,1698,1700,1723],{"className":1699},[884,885],[573,1701,1703,1720],{"className":1702},[889],[573,1704,1706],{"className":1705,"style":894},[893],[573,1707,1708,1711],{"style":897},[573,1709],{"className":1710,"style":902},[901],[573,1712,1714],{"className":1713},[906,907,908,909],[573,1715,1717],{"className":1716},[623,909],[573,1718,917],{"className":1719,"style":916},[623,624,909],[573,1721,922],{"className":1722},[921],[573,1724,1726],{"className":1725},[889],[573,1727,1729],{"className":1728,"style":929},[893],[573,1730],{},[573,1732],{"className":1733,"style":1137},[628],[573,1735,1103],{"className":1736},[1133],[573,1738],{"className":1739,"style":1137},[628],[573,1741,1743,1746],{"className":1742},[623],[573,1744,960],{"className":1745},[623,624],[573,1747,1749],{"className":1748},[880],[573,1750,1752,1775],{"className":1751},[884,885],[573,1753,1755,1772],{"className":1754},[889],[573,1756,1758],{"className":1757,"style":894},[893],[573,1759,1760,1763],{"style":897},[573,1761],{"className":1762,"style":902},[901],[573,1764,1766],{"className":1765},[906,907,908,909],[573,1767,1769],{"className":1768},[623,909],[573,1770,917],{"className":1771,"style":916},[623,624,909],[573,1773,922],{"className":1774},[921],[573,1776,1778],{"className":1777},[889],[573,1779,1781],{"className":1780,"style":929},[893],[573,1782],{}," ist ",[573,1785,1787,1805],{"className":1786},[576],[573,1788,1790],{"className":1789},[580],[582,1791,1792],{"xmlns":584},[586,1793,1794,1802],{},[589,1795,1796,1798,1800],{},[592,1797,1103],{},[596,1799,1106],{},[592,1801,960],{},[602,1803,1804],{"encoding":604},"\\det A",[573,1806,1808],{"className":1807,"ariaHidden":610},[609],[573,1809,1811,1815,1818,1821],{"className":1810},[614],[573,1812],{"className":1813,"style":1814},[618],"height:0.6944em;",[573,1816,1103],{"className":1817},[1133],[573,1819],{"className":1820,"style":1137},[628],[573,1822,960],{"className":1823},[623,624],". Der Ausdruck ",[573,1826,1828,1872],{"className":1827},[576],[573,1829,1831],{"className":1830},[580],[582,1832,1833],{"xmlns":584},[586,1834,1835,1869],{},[589,1836,1837,1839,1841,1843,1855,1857,1859],{},[596,1838,692],{"stretchy":691},[596,1840,697],{},[699,1842,701],{},[1204,1844,1845,1847],{},[596,1846,704],{"stretchy":691},[589,1848,1849,1851,1853],{},[592,1850,816],{},[596,1852,1214],{},[592,1854,857],{},[592,1856,1103],{},[596,1858,1106],{},[846,1860,1861,1863],{},[592,1862,960],{},[589,1864,1865,1867],{},[592,1866,816],{},[592,1868,857],{},[602,1870,1871],{"encoding":604},"(-1)^{i+j} \\det A_{ij}",[573,1873,1875],{"className":1874,"ariaHidden":610},[609],[573,1876,1878,1881,1884,1887,1890,1928,1931,1934,1937],{"className":1877},[614],[573,1879],{"className":1880,"style":1640},[618],[573,1882,692],{"className":1883},[733],[573,1885,697],{"className":1886},[623],[573,1888,701],{"className":1889},[623],[573,1891,1893,1896],{"className":1892},[758],[573,1894,704],{"className":1895},[758],[573,1897,1899],{"className":1898},[880],[573,1900,1902],{"className":1901},[884],[573,1903,1905],{"className":1904},[889],[573,1906,1908],{"className":1907,"style":1257},[893],[573,1909,1910,1913],{"style":1260},[573,1911],{"className":1912,"style":902},[901],[573,1914,1916],{"className":1915},[906,907,908,909],[573,1917,1919,1922,1925],{"className":1918},[623,909],[573,1920,816],{"className":1921},[623,624,909],[573,1923,1214],{"className":1924},[633,909],[573,1926,857],{"className":1927,"style":916},[623,624,909],[573,1929],{"className":1930,"style":1137},[628],[573,1932,1103],{"className":1933},[1133],[573,1935],{"className":1936,"style":1137},[628],[573,1938,1940,1943],{"className":1939},[623],[573,1941,960],{"className":1942},[623,624],[573,1944,1946],{"className":1945},[880],[573,1947,1949,1972],{"className":1948},[884,885],[573,1950,1952,1969],{"className":1951},[889],[573,1953,1955],{"className":1954,"style":894},[893],[573,1956,1957,1960],{"style":897},[573,1958],{"className":1959,"style":902},[901],[573,1961,1963],{"className":1962},[906,907,908,909],[573,1964,1966],{"className":1965},[623,909],[573,1967,917],{"className":1968,"style":916},[623,624,909],[573,1970,922],{"className":1971},[921],[573,1973,1975],{"className":1974},[889],[573,1976,1978],{"className":1977,"style":929},[893],[573,1979],{}," heißt ",[569,1982,1983],{},"Kofaktor",". Genauso darf man nach einer Spalte ",[573,1986,1988,2001],{"className":1987},[576],[573,1989,1991],{"className":1990},[580],[582,1992,1993],{"xmlns":584},[586,1994,1995,1999],{},[589,1996,1997],{},[592,1998,857],{},[602,2000,857],{"encoding":604},[573,2002,2004],{"className":2003,"ariaHidden":610},[609],[573,2005,2007,2010],{"className":2006},[614],[573,2008],{"className":2009,"style":1078},[618],[573,2011,857],{"className":2012,"style":916},[623,624]," entwickeln: ",[573,2015,2017,2096],{"className":2016},[576],[573,2018,2020],{"className":2019},[580],[582,2021,2022],{"xmlns":584},[586,2023,2024,2093],{},[589,2025,2026,2028,2030,2032,2035,2051,2053,2055,2057,2069,2079,2081,2083],{},[592,2027,1103],{},[596,2029,1106],{},[592,2031,960],{},[596,2033,2034],{},"=",[2036,2037,2038,2041,2049],"msubsup",{},[596,2039,2040],{},"∑",[589,2042,2043,2045,2047],{},[592,2044,816],{},[596,2046,2034],{},[699,2048,701],{},[592,2050,594],{},[596,2052,692],{"stretchy":691},[596,2054,697],{},[699,2056,701],{},[1204,2058,2059,2061],{},[596,2060,704],{"stretchy":691},[589,2062,2063,2065,2067],{},[592,2064,816],{},[596,2066,1214],{},[592,2068,857],{},[846,2070,2071,2073],{},[592,2072,850],{},[589,2074,2075,2077],{},[592,2076,816],{},[592,2078,857],{},[592,2080,1103],{},[596,2082,1106],{},[846,2084,2085,2087],{},[592,2086,960],{},[589,2088,2089,2091],{},[592,2090,816],{},[592,2092,857],{},[602,2094,2095],{"encoding":604},"\\det A = \\sum_{i=1}^{n} (-1)^{i+j} a_{ij} \\det 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Das Ergebnis hängt nicht von der Wahl der Zeile oder Spalte ab.",[565,2346,2347,2348,2376,2377,2412],{},"Am wenigsten Arbeit macht eine Zeile oder Spalte mit vielen Nullen, denn deren Summanden fallen weg. Oft lohnt es sich, vorher mit Zeilenumformungen Nullen zu erzeugen. Für große ",[573,2349,2351,2364],{"className":2350},[576],[573,2352,2354],{"className":2353},[580],[582,2355,2356],{"xmlns":584},[586,2357,2358,2362],{},[589,2359,2360],{},[592,2361,594],{},[602,2363,594],{"encoding":604},[573,2365,2367],{"className":2366,"ariaHidden":610},[609],[573,2368,2370,2373],{"className":2369},[614],[573,2371],{"className":2372,"style":643},[618],[573,2374,594],{"className":2375},[623,624]," ist der Aufwand ohne Nullen etwa ",[573,2378,2380,2397],{"className":2379},[576],[573,2381,2383],{"className":2382},[580],[582,2384,2385],{"xmlns":584},[586,2386,2387,2394],{},[589,2388,2389,2391],{},[592,2390,594],{},[596,2392,2393],{"stretchy":691},"!",[602,2395,2396],{"encoding":604},"n!",[573,2398,2400],{"className":2399,"ariaHidden":610},[609],[573,2401,2403,2406,2409],{"className":2402},[614],[573,2404],{"className":2405,"style":1814},[618],[573,2407,594],{"className":2408},[623,624],[573,2410,2393],{"className":2411},[758],", dann ist die Umformung auf Dreiecksform 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2 enthält nur einen Eintrag ungleich ",[573,2986,2988,3001],{"className":2987},[576],[573,2989,2991],{"className":2990},[580],[582,2992,2993],{"xmlns":584},[586,2994,2995,2999],{},[589,2996,2997],{},[699,2998,2453],{},[602,3000,2453],{"encoding":604},[573,3002,3004],{"className":3003,"ariaHidden":610},[609],[573,3005,3007,3011],{"className":3006},[614],[573,3008],{"className":3009,"style":3010},[618],"height:0.6444em;",[573,3012,2453],{"className":3013},[623],", nämlich ",[573,3016,3018,3041],{"className":3017},[576],[573,3019,3021],{"className":3020},[580],[582,3022,3023],{"xmlns":584},[586,3024,3025,3038],{},[589,3026,3027,3034,3036],{},[846,3028,3029,3031],{},[592,3030,850],{},[699,3032,3033],{},"23",[596,3035,2034],{},[699,3037,701],{},[602,3039,3040],{"encoding":604},"a_{23} = 1",[573,3042,3044,3105],{"className":3043,"ariaHidden":610},[609],[573,3045,3047,3051,3096,3099,3102],{"className":3046},[614],[573,3048],{"className":3049,"style":3050},[618],"height:0.5806em;vertical-align:-0.15em;",[573,3052,3054,3057],{"className":3053},[623],[573,3055,850],{"className":3056},[623,624],[573,3058,3060],{"className":3059},[880],[573,3061,3063,3087],{"className":3062},[884,885],[573,3064,3066,3084],{"className":3065},[889],[573,3067,3070],{"className":3068,"style":3069},[893],"height:0.3011em;",[573,3071,3072,3075],{"style":897},[573,3073],{"className":3074,"style":902},[901],[573,3076,3078],{"className":3077},[906,907,908,909],[573,3079,3081],{"className":3080},[623,909],[573,3082,3033],{"className":3083},[623,909],[573,3085,922],{"className":3086},[921],[573,3088,3090],{"className":3089},[889],[573,3091,3094],{"className":3092,"style":3093},[893],"height:0.15em;",[573,3095],{},[573,3097],{"className":3098,"style":2117},[628],[573,3100,2034],{"className":3101},[2121],[573,3103],{"className":3104,"style":2117},[628],[573,3106,3108,3111],{"className":3107},[614],[573,3109],{"className":3110,"style":3010},[618],[573,3112,701],{"className":3113},[623],". Das Vorzeichen ist ",[573,3116,3118,3154],{"className":3117},[576],[573,3119,3121],{"className":3120},[580],[582,3122,3123],{"xmlns":584},[586,3124,3125,3151],{},[589,3126,3127,3129,3131,3133,3145,3147,3149],{},[596,3128,692],{"stretchy":691},[596,3130,697],{},[699,3132,701],{},[1204,3134,3135,3137],{},[596,3136,704],{"stretchy":691},[589,3138,3139,3141,3143],{},[699,3140,2456],{},[596,3142,1214],{},[699,3144,2475],{},[596,3146,2034],{},[596,3148,697],{},[699,3150,701],{},[602,3152,3153],{"encoding":604},"(-1)^{2+3} = -1",[573,3155,3157,3221],{"className":3156,"ariaHidden":610},[609],[573,3158,3160,3164,3167,3170,3173,3212,3215,3218],{"className":3159},[614],[573,3161],{"className":3162,"style":3163},[618],"height:1.0641em;vertical-align:-0.25em;",[573,3165,692],{"className":3166},[733],[573,3168,697],{"className":3169},[623],[573,3171,701],{"className":3172},[623],[573,3174,3176,3179],{"className":3175},[758],[573,3177,704],{"className":3178},[758],[573,3180,3182],{"className":3181},[880],[573,3183,3185],{"className":3184},[884],[573,3186,3188],{"className":3187},[889],[573,3189,3192],{"className":3190,"style":3191},[893],"height:0.8141em;",[573,3193,3194,3197],{"style":1260},[573,3195],{"className":3196,"style":902},[901],[573,3198,3200],{"className":3199},[906,907,908,909],[573,3201,3203,3206,3209],{"className":3202},[623,909],[573,3204,2456],{"className":3205},[623,909],[573,3207,1214],{"className":3208},[633,909],[573,3210,2475],{"className":3211},[623,909],[573,3213],{"className":3214,"style":2117},[628],[573,3216,2034],{"className":3217},[2121],[573,3219],{"className":3220,"style":2117},[628],[573,3222,3224,3228,3231],{"className":3223},[614],[573,3225],{"className":3226,"style":3227},[618],"height:0.7278em;vertical-align:-0.0833em;",[573,3229,697],{"className":3230},[623],[573,3232,701],{"className":3233},[623],". Streichen von Zeile 2 und Spalte 3 lässt",[573,3236,3238],{"className":3237},[2420],[573,3239,3241,3392],{"className":3240},[576],[573,3242,3244],{"className":3243},[580],[582,3245,3246],{"xmlns":584,"display":2429},[586,3247,3248,3389],{},[589,3249,3250,3256,3258,3327,3329,3332,3334,3336,3342,3344,3346,3348,3350,3352,3354,3356,3358,3360,3362,3364,3366,3368,3370,3372,3374,3376,3378,3380,3382,3384,3386],{},[846,3251,3252,3254],{},[592,3253,960],{},[699,3255,3033],{},[596,3257,2034],{},[589,3259,3260,3262,3325],{},[596,3261,692],{"fence":610},[1296,3263,3265,3285,3305],{"rowspacing":2444,"columnalign":3264,"columnspacing":2446},"center center center",[1301,3266,3267,3273,3279],{},[1304,3268,3269],{},[1293,3270,3271],{"scriptlevel":2453,"displaystyle":691},[699,3272,2456],{},[1304,3274,3275],{},[1293,3276,3277],{"scriptlevel":2453,"displaystyle":691},[699,3278,701],{},[1304,3280,3281],{},[1293,3282,3283],{"scriptlevel":2453,"displaystyle":691},[699,3284,2475],{},[1301,3286,3287,3293,3299],{},[1304,3288,3289],{},[1293,3290,3291],{"scriptlevel":2453,"displaystyle":691},[699,3292,701],{},[1304,3294,3295],{},[1293,3296,3297],{"scriptlevel":2453,"displaystyle":691},[699,3298,2516],{},[1304,3300,3301],{},[1293,3302,3303],{"scriptlevel":2453,"displaystyle":691},[699,3304,2456],{},[1301,3306,3307,3313,3319],{},[1304,3308,3309],{},[1293,3310,3311],{"scriptlevel":2453,"displaystyle":691},[699,3312,2453],{},[1304,3314,3315],{},[1293,3316,3317],{"scriptlevel":2453,"displaystyle":691},[699,3318,701],{},[1304,3320,3321],{},[1293,3322,3323],{"scriptlevel":2453,"displaystyle":691},[699,3324,701],{},[596,3326,704],{"fence":610},[596,3328,1181],{"separator":610},[628,3330],{"width":3331},"2em",[592,3333,1103],{},[596,3335,1106],{},[846,3337,3338,3340],{},[592,3339,960],{},[699,3341,3033],{},[596,3343,2034],{},[699,3345,2456],{},[596,3347,692],{"stretchy":691},[699,3349,2516],{},[596,3351,697],{},[699,3353,2456],{},[596,3355,704],{"stretchy":691},[596,3357,697],{},[699,3359,701],{},[596,3361,692],{"stretchy":691},[699,3363,701],{},[596,3365,697],{},[699,3367,2453],{},[596,3369,704],{"stretchy":691},[596,3371,1214],{},[699,3373,2475],{},[596,3375,692],{"stretchy":691},[699,3377,701],{},[596,3379,697],{},[699,3381,2453],{},[596,3383,704],{"stretchy":691},[596,3385,2034],{},[699,3387,3388],{},"6.",[602,3390,3391],{"encoding":604},"A_{23} = \\begin{pmatrix} 2 & 1 & 3 \\\\ 1 & 4 & 2 \\\\ 0 & 1 & 1 \\end{pmatrix}, \\qquad \\det A_{23} = 2(4 - 2) - 1(1 - 0) + 3(1 - 0) = 6.",[573,3393,3395,3454,3808,3832,3853,3877,3898,3922,3943],{"className":3394,"ariaHidden":610},[609],[573,3396,3398,3402,3445,3448,3451],{"className":3397},[614],[573,3399],{"className":3400,"style":3401},[618],"height:0.8333em;vertical-align:-0.15em;",[573,3403,3405,3408],{"className":3404},[623],[573,3406,960],{"className":3407},[623,624],[573,3409,3411],{"className":3410},[880],[573,3412,3414,3437],{"className":3413},[884,885],[573,3415,3417,3434],{"className":3416},[889],[573,3418,3420],{"className":3419,"style":3069},[893],[573,3421,3422,3425],{"style":897},[573,3423],{"className":3424,"style":902},[901],[573,3426,3428],{"className":3427},[906,907,908,909],[573,3429,3431],{"className":3430},[623,909],[573,3432,3033],{"className":3433},[623,909],[573,3435,922],{"className":3436},[921],[573,3438,3440],{"className":3439},[889],[573,3441,3443],{"className":3442,"style":3093},[893],[573,3444],{},[573,3446],{"className":3447,"style":2117},[628],[573,3449,2034],{"className":3450},[2121],[573,3452],{"className":3453,"style":2117},[628],[573,3455,3457,3461,3736,3739,3743,3747,3750,3753,3756,3799,3802,3805],{"className":3456},[614],[573,3458],{"className":3459,"style":3460},[618],"height:3.6em;vertical-align:-1.55em;",[573,3462,3464,3509,3698],{"className":3463},[2594],[573,3465,3467],{"className":3466},[733],[573,3468,3470],{"className":3469},[2601,2602],[573,3471,3473,3500],{"className":3472},[884,885],[573,3474,3476,3497],{"className":3475},[889],[573,3477,3480],{"className":3478,"style":3479},[893],"height:2.05em;",[573,3481,3483,3487],{"style":3482},"top:-4.05em;",[573,3484],{"className":3485,"style":3486},[901],"height:5.6em;",[573,3488,3490],{"style":3489},"width:0.875em;height:3.6em;",[2624,3491,3494],{"xmlns":2626,"width":2627,"height":3492,"viewBox":3493},"3.6em","0 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-6",[573,3999,4001,4025,4046,4064,4082],{"className":4000,"ariaHidden":610},[609],[573,4002,4004,4007,4010,4013,4016,4019,4022],{"className":4003},[614],[573,4005],{"className":4006,"style":1814},[618],[573,4008,1103],{"className":4009},[1133],[573,4011],{"className":4012,"style":1137},[628],[573,4014,960],{"className":4015},[623,624],[573,4017],{"className":4018,"style":2117},[628],[573,4020,2034],{"className":4021},[2121],[573,4023],{"className":4024,"style":2117},[628],[573,4026,4028,4031,4034,4037,4040,4043],{"className":4027},[614],[573,4029],{"className":4030,"style":3227},[618],[573,4032,697],{"className":4033},[623],[573,4035,701],{"className":4036},[623],[573,4038],{"className":4039,"style":629},[628],[573,4041,3981],{"className":4042},[633],[573,4044],{"className":4045,"style":629},[628],[573,4047,4049,4052,4055,4058,4061],{"className":4048},[614],[573,4050],{"className":4051,"style":3010},[618],[573,4053,701],{"className":4054},[623],[573,4056],{"className":4057,"style":629},[628],[573,4059,3981],{"className":4060},[633],[573,4062],{"className":4063,"style":629},[628],[573,4065,4067,4070,4073,4076,4079],{"className":4066},[614],[573,4068],{"className":4069,"style":3010},[618],[573,4071,3988],{"className":4072},[623],[573,4074],{"className":4075,"style":2117},[628],[573,4077,2034],{"className":4078},[2121],[573,4080],{"className":4081,"style":2117},[628],[573,4083,4085,4088,4091],{"className":4084},[614],[573,4086],{"className":4087,"style":3227},[618],[573,4089,697],{"className":4090},[623],[573,4092,3988],{"className":4093},[623],". Die ",[573,4096,4098,4116],{"className":4097},[576],[573,4099,4101],{"className":4100},[580],[582,4102,4103],{"xmlns":584},[586,4104,4105,4113],{},[589,4106,4107,4109,4111],{},[699,4108,2475],{},[596,4110,598],{},[699,4112,2475],{},[602,4114,4115],{"encoding":604},"3 \\times 3",[573,4117,4119,4137],{"className":4118,"ariaHidden":610},[609],[573,4120,4122,4125,4128,4131,4134],{"className":4121},[614],[573,4123],{"className":4124,"style":3227},[618],[573,4126,2475],{"className":4127},[623],[573,4129],{"className":4130,"style":629},[628],[573,4132,598],{"className":4133},[633],[573,4135],{"className":4136,"style":629},[628],[573,4138,4140,4143],{"className":4139},[614],[573,4141],{"className":4142,"style":3010},[618],[573,4144,2475],{"className":4145},[623],"-Determinante wurde dabei selbst nach Zeile 1 entwickelt.",[560,4148,4150],{"id":4149},"typische-fehler","Typische Fehler",[4152,4153,4154,4350,4430,4433],"ul",{},[937,4155,4156,4157,4245,4246,4319,4320,4349],{},"Das Schachbrettvorzeichen vergessen oder nach der Position in der Summe statt nach ",[573,4158,4160,4189],{"className":4159},[576],[573,4161,4163],{"className":4162},[580],[582,4164,4165],{"xmlns":584},[586,4166,4167,4187],{},[589,4168,4169,4171,4173,4175],{},[596,4170,692],{"stretchy":691},[596,4172,697],{},[699,4174,701],{},[1204,4176,4177,4179],{},[596,4178,704],{"stretchy":691},[589,4180,4181,4183,4185],{},[592,4182,816],{},[596,4184,1214],{},[592,4186,857],{},[602,4188,1219],{"encoding":604},[573,4190,4192],{"className":4191,"ariaHidden":610},[609],[573,4193,4195,4198,4201,4204,4207],{"className":4194},[614],[573,4196],{"className":4197,"style":1229},[618],[573,4199,692],{"className":4200},[733],[573,4202,697],{"className":4203},[623],[573,4205,701],{"className":4206},[623],[573,4208,4210,4213],{"className":4209},[758],[573,4211,704],{"className":4212},[758],[573,4214,4216],{"className":4215},[880],[573,4217,4219],{"className":4218},[884],[573,4220,4222],{"className":4221},[889],[573,4223,4225],{"className":4224,"style":1257},[893],[573,4226,4227,4230],{"style":1260},[573,4228],{"className":4229,"style":902},[901],[573,4231,4233],{"className":4232},[906,907,908,909],[573,4234,4236,4239,4242],{"className":4235},[623,909],[573,4237,816],{"className":4238},[623,624,909],[573,4240,1214],{"className":4241},[633,909],[573,4243,857],{"className":4244,"style":916},[623,624,909]," vergeben. Bei ",[573,4247,4249,4267],{"className":4248},[576],[573,4250,4252],{"className":4251},[580],[582,4253,4254],{"xmlns":584},[586,4255,4256,4264],{},[589,4257,4258],{},[846,4259,4260,4262],{},[592,4261,850],{},[699,4263,3033],{},[602,4265,4266],{"encoding":604},"a_{23}",[573,4268,4270],{"className":4269,"ariaHidden":610},[609],[573,4271,4273,4276],{"className":4272},[614],[573,4274],{"className":4275,"style":3050},[618],[573,4277,4279,4282],{"className":4278},[623],[573,4280,850],{"className":4281},[623,624],[573,4283,4285],{"className":4284},[880],[573,4286,4288,4311],{"className":4287},[884,885],[573,4289,4291,4308],{"className":4290},[889],[573,4292,4294],{"className":4293,"style":3069},[893],[573,4295,4296,4299],{"style":897},[573,4297],{"className":4298,"style":902},[901],[573,4300,4302],{"className":4301},[906,907,908,909],[573,4303,4305],{"className":4304},[623,909],[573,4306,3033],{"className":4307},[623,909],[573,4309,922],{"className":4310},[921],[573,4312,4314],{"className":4313},[889],[573,4315,4317],{"className":4316,"style":3093},[893],[573,4318],{}," ist es ",[573,4321,4323,4337],{"className":4322},[576],[573,4324,4326],{"className":4325},[580],[582,4327,4328],{"xmlns":584},[586,4329,4330,4334],{},[589,4331,4332],{},[596,4333,697],{},[602,4335,4336],{"encoding":604},"-",[573,4338,4340],{"className":4339,"ariaHidden":610},[609],[573,4341,4343,4346],{"className":4342},[614],[573,4344],{"className":4345,"style":619},[618],[573,4347,697],{"className":4348},[623],", auch wenn es der einzige Summand ist.",[937,4351,4352,4353,4429],{},"Den Eintrag ",[573,4354,4356,4377],{"className":4355},[576],[573,4357,4359],{"className":4358},[580],[582,4360,4361],{"xmlns":584},[586,4362,4363,4375],{},[589,4364,4365],{},[846,4366,4367,4369],{},[592,4368,850],{},[589,4370,4371,4373],{},[592,4372,816],{},[592,4374,857],{},[602,4376,860],{"encoding":604},[573,4378,4380],{"className":4379,"ariaHidden":610},[609],[573,4381,4383,4386],{"className":4382},[614],[573,4384],{"className":4385,"style":870},[618],[573,4387,4389,4392],{"className":4388},[623],[573,4390,850],{"className":4391},[623,624],[573,4393,4395],{"className":4394},[880],[573,4396,4398,4421],{"className":4397},[884,885],[573,4399,4401,4418],{"className":4400},[889],[573,4402,4404],{"className":4403,"style":894},[893],[573,4405,4406,4409],{"style":897},[573,4407],{"className":4408,"style":902},[901],[573,4410,4412],{"className":4411},[906,907,908,909],[573,4413,4415],{"className":4414},[623,909],[573,4416,917],{"className":4417,"style":916},[623,624,909],[573,4419,922],{"className":4420},[921],[573,4422,4424],{"className":4423},[889],[573,4425,4427],{"className":4426,"style":929},[893],[573,4428],{}," vergessen und nur Vorzeichen mal Minor addieren.",[937,4431,4432],{},"Beim Streichen die falsche Spalte entfernen, besonders bei Entwicklung nach einer Spalte.",[937,4434,4435,4436,4487,4488,1570],{},"Die Regel von Sarrus auf ",[573,4437,4439,4457],{"className":4438},[576],[573,4440,4442],{"className":4441},[580],[582,4443,4444],{"xmlns":584},[586,4445,4446,4454],{},[589,4447,4448,4450,4452],{},[699,4449,2516],{},[596,4451,598],{},[699,4453,2516],{},[602,4455,4456],{"encoding":604},"4 \\times 4",[573,4458,4460,4478],{"className":4459,"ariaHidden":610},[609],[573,4461,4463,4466,4469,4472,4475],{"className":4462},[614],[573,4464],{"className":4465,"style":3227},[618],[573,4467,2516],{"className":4468},[623],[573,4470],{"className":4471,"style":629},[628],[573,4473,598],{"className":4474},[633],[573,4476],{"className":4477,"style":629},[628],[573,4479,4481,4484],{"className":4480},[614],[573,4482],{"className":4483,"style":3010},[618],[573,4485,2516],{"className":4486},[623],"-Matrizen übertragen. Sie gilt nur für ",[573,4489,4491,4508],{"className":4490},[576],[573,4492,4494],{"className":4493},[580],[582,4495,4496],{"xmlns":584},[586,4497,4498,4506],{},[589,4499,4500,4502,4504],{},[699,4501,2475],{},[596,4503,598],{},[699,4505,2475],{},[602,4507,4115],{"encoding":604},[573,4509,4511,4529],{"className":4510,"ariaHidden":610},[609],[573,4512,4514,4517,4520,4523,4526],{"className":4513},[614],[573,4515],{"className":4516,"style":3227},[618],[573,4518,2475],{"className":4519},[623],[573,4521],{"className":4522,"style":629},[628],[573,4524,598],{"className":4525},[633],[573,4527],{"className":4528,"style":629},[628],[573,4530,4532,4535],{"className":4531},[614],[573,4533],{"className":4534,"style":3010},[618],[573,4536,2475],{"className":4537},[623],{"title":4539,"searchDepth":4540,"depth":4540,"links":4541},"",2,[4542,4543,4544],{"id":562,"depth":4540,"text":563},{"id":2415,"depth":4540,"text":2416},{"id":4149,"depth":4540,"text":4150},[4546,4549,4551,4553,4555,4558],{"lhs":4547,"rhs":4548},"a*det([[f, g], [k, l]]) - b*det([[d, g], [h, l]]) + c*det([[d, f], [h, k]])","det([[a, b, c], [d, f, g], [h, k, l]])",{"lhs":4550,"rhs":4548},"-b*det([[d, g], [h, l]]) + f*det([[a, c], [h, l]]) - k*det([[a, c], [d, g]])",{"lhs":4552,"rhs":3988},"det([[2, 1, 3], [1, 4, 2], [0, 1, 1]])",{"lhs":4554,"rhs":3988},"2*(4*1 - 2*1) - 1*(1*1 - 2*0) + 3*(1*1 - 4*0)",{"lhs":4556,"rhs":4557},"(-1)^(2 + 3) * 1 * 6","-6",{"lhs":4559,"rhs":4557},"det([[2, 1, 0, 3], [0, 0, 1, 0], [1, 4, 5, 2], [0, 1, 7, 1]])","A \\in \\mathbb{R}^{n \\times n},\\ i \\in \\{1, \\ldots, n\\} \\text{ fest}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>×\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A \\in \\mathbb{R}^{n \\times n},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9658em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">×\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmo stretchy=\"false\">{\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003Cmo stretchy=\"false\">}\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i \\in \\{1, \\ldots, n\\}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6986em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">{\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003Cspan class=\"mclose\">}\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">fest\u003C\u002Fspan>",null,"md","laplace-expansion","\\det A = \\sum_{j=1}^{n} (-1)^{i+j}\\, a_{ij} \\det A_{ij}","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi>det\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>j\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmtext> \u003C\u002Fmtext>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmi>det\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>A\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\det A = \\sum_{j=1}^{n} (-1)^{i+j}\\, a_{ij} \\det A_{ij}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">det\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:3.0652em;vertical-align:-1.4138em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6514em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.4138em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8747em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">ij\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">det\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">ij\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Flaplace-entwicklung",[],[4573,4574,4575],"determinant-3x3","determinant-rules","adjugate-inverse",{"title":548,"description":4539},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Flaplace-entwicklung","Die Determinante ist die Summe über eine Zeile aus Eintrag mal Vorzeichen mal Determinante der Streichungsmatrix.",[4580,4581,4582,4583,4584],"determinante","laplace","entwicklungssatz","minor","kofaktor",[4586,4589,4592,4595,4598,4601,4604],{"symbol":960,"meaning":4587,"symbolHtml":4588},"quadratische n × n-Matrix","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":860,"meaning":4590,"symbolHtml":4591},"Eintrag von A in Zeile i und Spalte j","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a_{ij}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7167em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">ij\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":969,"meaning":4593,"symbolHtml":4594},"(n − 1) × (n − 1)-Matrix, die aus A durch Streichen von Zeile i und Spalte j entsteht","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>A\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A_{ij}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9694em;vertical-align:-0.2861em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">ij\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2861em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":594,"meaning":4596,"symbolHtml":4597},"Zeilen- und Spaltenzahl von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":857,"meaning":4599,"symbolHtml":4600},"Spaltenindex, über den summiert wird","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">j\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.854em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":816,"meaning":4602,"symbolHtml":4603},"Nummer der Zeile, nach der entwickelt wird","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">i\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6595em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">i\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":1219,"meaning":4605,"symbolHtml":4606},"Vorzeichen nach dem Schachbrettmuster","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmo>−\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>j\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(-1)^{i+j}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.0747em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8247em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0572em;\">j\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","efjd6G8W6QMGLz178_b94Bc9qTF5hxb0hIxC7T5Biro",[4609,4612,4615],{"stem":4610,"title":4611,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fdeterminante-3x3","Determinante einer 3×3-Matrix (Regel von Sarrus)",{"stem":4613,"title":4614,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Finverse-ueber-adjunkte","Inverse über die Adjunkte",{"stem":4616,"title":4617,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Frechenregeln-determinanten","Rechenregeln für Determinanten",1793824278803]