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1、线性代数第一章 行列式b a -a ba a - a a11 22 12 21定义：设 a11,a12,a21,a22记a a - a a 为11 22 12 21aiia21a12a22aa称1112为二阶行列式aa21221 二阶和三阶行列式元一次线性方程组与二阶行列式结论：如果a a - a a丰0，则二元线性方程组11 22 12 21a x + a x = b11 112 21a x + a x = b21 122 22的解为a b -bax = 11241。2 a b -ba11 2 1 21a11a12a13如果D =aaa丰0,212223aaa313233定理：有了行列式的符
2、号，二元线性方程组的求解公式可以改写为V -a11a122AJ2aa1112aa2122)a112a222aa1112aa2122、三阶行列式与三元一次线性方程组a11a12a13定义：a21a22a23aaa313233=aaa+ a aa + a a a - a a a - a a a - a a a11 22331223 3113 21 3213 22 3112 21 3311 23 32则（x*, x2,x；）是下面的三元线性方程组的解ax+ ax+ ax= b1111221331ax+ ax+ ax= b2112222332ax+ ax+ ax= b3113223333aa1213x
3、*=aa/ D , x* =1222232baa33233aaa111213其中aaa为系数行列式。212223aaa313233当且仅当证明：略。abaa ab1111311 12 1aba/ D, x* =a ab/ D21223321 22 2abaaab3133331323性质 1：行列式行列互换，其值不变。a11a12a13a11a21a31即aaa=aaa212223122232aaaaaa313233132333性质 2：行列式某两行或列互换，其值变号。例如aaaaaa111213212223aaa=aaa212223111213aaaaaa313233313233推论：行列式有
4、两行相同，其值为零。性质3：行列式某一行的所有数乘一常数等于行列式乘该常数。例如aaaaaa111213111213kakaka=kaaa212223212223aaaaaa313233313233推论：行列式某一行或列的公因数可以提到行列式外面。推论：行列式有一行全为零，其值为零。性质 4：行列式有两行成比例时，其值为零。性质 5：行列式关于它的每一行和每一列都是线性的。例如aaaaaaaaa111213111213111213a + ba + ba + b=aaa+bbb21 2122 222323212223212223aaaaaaaaa313233313233313233，性质6：将行
5、列式的某一行（列）的所有元素都乘以数k后加到另一行（列）对应位置的元素上, 其值不变。例如aaaaaa111213111213a + kaa + kaa + ka=aaa21 1122 122313212223aaaaaa313233313233性质 7：行列式按某一行展开aaa111213aaaaaaaaa=a2223a2123+ a212221222311aa12aa13aaaaa323331333132313233aa=b定理的证明：用2223乘第一个方程ax + ax+ ax得aa11112 213313233aaaaaabaaa2223 X+ a2223 x + a2223x =22
6、2311aa112aa 213aa31aa3233323332333233aab用1213乘第一个方程 a x+ a x + ax =得aa21 122 223 323233aaaaaabaaa1213 X+ a1213 x + a1213x =121321aa122aa223aa32aa3233323332333233同理，有aaaaaa=baaa1213x + a1213x + a1213x1213。31aa132aa233aa33aa2223222322232223+ （-1）+，得aaaaaa(a2223a1213+ a1213)x11aa21aa31aa1323332332223aa
7、aaaa+(a2223a1213+ a1213)x12aa22aa32aa2323332332223aaaaaa+(a2223a1213+ a1213)x13aa23aa33aa3323332332223aaaaaa=b 2223-b1213+ b12131 aa2 aa3aa323332332223利用性质 7，得a aaaaaaaabaa11121312121313121311213a aax +aaax +aaax =baa21222312222232232223322223a aaaaaaaabaa31323332323333323333233从而aaabaa11121311213aa
8、ax=baa212223122223aaabaa31323333233Iax + ax + ax=011 1 12 2 13 3定理：a x + a x + a x = 0有非零解当且仅当系数行列式D二0。21 1 22 2 23 3ax + ax + ax = 031 1 32 2 33 3证明：必要性：若齐次方程组有非零解，如果D鼻0，由前面的定理，矛盾。a22aa23aa21aa23aa21aa22）带入第2和第3个方程，容易验证它是方程组的解。a充分性：若D = 0，注意a11a12a13aaaaaaaaa=a 2223 + a -2123+a212221222311 aa12aa13aaaaa32333133丿3132313233aaaaaa因此如果
