2.3.1 行列式的定义与性质

n阶方阵的行列式记为或,称为n阶行列式。计算公式为

其中为整数的一个排列，是这n个整数所有排列的集合，显然有种排列。%3C%2Ftitle%3E%0A%3Cdefs%20aria-hidden%3D%22true%22%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-3C4%22%20d%3D%22M39%20284Q18%20284%2018%20294Q18%20301%2045%20338T99%20398Q134%20425%20164%20429Q170%20431%20332%20431Q492%20431%20497%20429Q517%20424%20517%20402Q517%20388%20508%20376T485%20360Q479%20358%20389%20358T299%20356Q298%20355%20283%20274T251%20109T233%2020Q228%205%20215%20-4T186%20-13Q153%20-13%20153%2020V30L203%20192Q214%20228%20227%20272T248%20336L254%20357Q254%20358%20208%20358Q206%20358%20197%20358T183%20359Q105%20359%2061%20295Q56%20287%2053%20286T39%20284Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-28%22%20d%3D%22M94%20250Q94%20319%20104%20381T127%20488T164%20576T202%20643T244%20695T277%20729T302%20750H315H319Q333%20750%20333%20741Q333%20738%20316%20720T275%20667T226%20581T184%20443T167%20250T184%2058T225%20-81T274%20-167T316%20-220T333%20-241Q333%20-250%20318%20-250H315H302L274%20-226Q180%20-141%20137%20-14T94%20250Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-6A%22%20d%3D%22M297%20596Q297%20627%20318%20644T361%20661Q378%20661%20389%20651T403%20623Q403%20595%20384%20576T340%20557Q322%20557%20310%20567T297%20596ZM288%20376Q288%20405%20262%20405Q240%20405%20220%20393T185%20362T161%20325T144%20293L137%20279Q135%20278%20121%20278H107Q101%20284%20101%20286T105%20299Q126%20348%20164%20391T252%20441Q253%20441%20260%20441T272%20442Q296%20441%20316%20432Q341%20418%20354%20401T367%20348V332L318%20133Q267%20-67%20264%20-75Q246%20-125%20194%20-164T75%20-204Q25%20-204%207%20-183T-12%20-137Q-12%20-110%207%20-91T53%20-71Q70%20-71%2082%20-81T95%20-112Q95%20-148%2063%20-167Q69%20-168%2077%20-168Q111%20-168%20139%20-140T182%20-74L193%20-32Q204%2011%20219%2072T251%20197T278%20308T289%20365Q289%20372%20288%20376Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-31%22%20d%3D%22M213%20578L200%20573Q186%20568%20160%20563T102%20556H83V602H102Q149%20604%20189%20617T245%20641T273%20663Q275%20666%20285%20666Q294%20666%20302%20660V361L303%2061Q310%2054%20315%2052T339%2048T401%2046H427V0H416Q395%203%20257%203Q121%203%20100%200H88V46H114Q136%2046%20152%2046T177%2047T193%2050T201%2052T207%2057T213%2061V578Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-32%22%20d%3D%22M109%20429Q82%20429%2066%20447T50%20491Q50%20562%20103%20614T235%20666Q326%20666%20387%20610T449%20465Q449%20422%20429%20383T381%20315T301%20241Q265%20210%20201%20149L142%2093L218%2092Q375%2092%20385%2097Q392%2099%20409%20186V189H449V186Q448%20183%20436%2095T421%203V0H50V19V31Q50%2038%2056%2046T86%2081Q115%20113%20136%20137Q145%20147%20170%20174T204%20211T233%20244T261%20278T284%20308T305%20340T320%20369T333%20401T340%20431T343%20464Q343%20527%20309%20573T212%20619Q179%20619%20154%20602T119%20569T109%20550Q109%20549%20114%20549Q132%20549%20151%20535T170%20489Q170%20464%20154%20447T109%20429Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-2026%22%20d%3D%22M78%2060Q78%2084%2095%20102T138%20120Q162%20120%20180%20104T199%2061Q199%2036%20182%2018T139%200T96%2017T78%2060ZM525%2060Q525%2084%20542%20102T585%20120Q609%20120%20627%20104T646%2061Q646%2036%20629%2018T586%200T543%2017T525%2060ZM972%2060Q972%2084%20989%20102T1032%20120Q1056%20120%201074%20104T1093%2061Q1093%2036%201076%2018T1033%200T990%2017T972%2060Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMATHI-6E%22%20d%3D%22M21%20287Q22%20293%2024%20303T36%20341T56%20388T89%20425T135%20442Q171%20442%20195%20424T225%20390T231%20369Q231%20367%20232%20367L243%20378Q304%20442%20382%20442Q436%20442%20469%20415T503%20336T465%20179T427%2052Q427%2026%20444%2026Q450%2026%20453%2027Q482%2032%20505%2065T540%20145Q542%20153%20560%20153Q580%20153%20580%20145Q580%20144%20576%20130Q568%20101%20554%2073T508%2017T439%20-10Q392%20-10%20371%2017T350%2073Q350%2092%20386%20193T423%20345Q423%20404%20379%20404H374Q288%20404%20229%20303L222%20291L189%20157Q156%2026%20151%2016Q138%20-11%20108%20-11Q95%20-11%2087%20-5T76%207T74%2017Q74%2030%20112%20180T152%20343Q153%20348%20153%20366Q153%20405%20129%20405Q91%20405%2066%20305Q60%20285%2060%20284Q58%20278%2041%20278H27Q21%20284%2021%20287Z%22%3E%3C%2Fpath%3E%0A%3Cpath%20stroke-width%3D%221%22%20id%3D%22E1-MJMAIN-29%22%20d%3D%22M60%20749L64%20750Q69%20750%2074%20750H86L114%20726Q208%20641%20251%20514T294%20250Q294%20182%20284%20119T261%2012T224%20-76T186%20-143T145%20-194T113%20-227T90%20-246Q87%20-249%2086%20-250H74Q66%20-250%2063%20-250T58%20-247T55%20-238Q56%20-237%2066%20-225Q221%20-64%20221%20250T66%20725Q56%20737%2055%20738Q55%20746%2060%20749Z%22%3E%3C%2Fpath%3E%0A%3C%2Fdefs%3E%0A%3Cg%20stroke%3D%22currentColor%22%20fill%3D%22currentColor%22%20stroke-width%3D%220%22%20transform%3D%22matrix(1%200%200%20-1%200%200)%22%20aria-hidden%3D%22true%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-3C4%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-28%22%20x%3D%22517%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(907%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6A%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-31%22%20x%3D%22583%22%20y%3D%22-213%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3Cg%20transform%3D%22translate(1773%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6A%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMAIN-32%22%20x%3D%22583%22%20y%3D%22-213%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-2026%22%20x%3D%222806%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3Cg%20transform%3D%22translate(4145%2C0)%22%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMATHI-6A%22%20x%3D%220%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%20%3Cuse%20transform%3D%22scale(0.707)%22%20xlink%3Ahref%3D%22%23E1-MJMATHI-6E%22%20x%3D%22583%22%20y%3D%22-213%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%20%3Cuse%20xlink%3Ahref%3D%22%23E1-MJMAIN-29%22%20x%3D%225082%22%20y%3D%220%22%3E%3C%2Fuse%3E%0A%3C%2Fg%3E%0A%3C%2Fsvg%3E#card=math&code=%5Ctau%28j1j_2%20%5Cdots%20j_n%29&id=XwEa0)为排列的逆序数，对于一个排列，如果,但，则称为一个逆序，排列中所有逆序的数量称为排列的逆序数。
按照定义，一个行列式可以按照行或列进行递归展开，称为拉普拉斯展开
其中是去掉矩阵的第行第列之后的阶矩阵的行列式，并且带有符号,为行号和列号之后。称为的代数余子式，不带符号的子行列式称为余子式。
根据式（2.28）的定义，如果一个矩阵为对角矩阵，则其行列式为矩阵的主对角线元素的乘积。这是因为个求和项中，除全部主对角线元素构成的项之外，其他的项的乘积中都含有0.
上下三角矩阵的行列式为其主对角线元素的乘积。
下面介绍行列式的若干重要性质。行列式具有多线性，可以按照某一行或列的线性组合拆分成两个行列式之和.
如果行列式的两行或列相等，那么行列式的值为0.
如果行列式的某行元素乘k，则行列式变为之前的k倍。
如果 行列式的两行交换，行列式反号。