[|A|= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix} ]

则(|A|)具有以下性质：

5.1 性质1：(|A|^T=|A|)

性质1的证明：

由矩阵转置相关性质可知：

[(a_{ij})^T=a(_{ji})\ ]

而：

[|A|=sum (-1)^ta_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...cdot a_{np_n}\ |A|^T=sum (-1)^ta_{p_11}cdot a_{p_22}cdot a_{p_33}cdot ... a_{p_nn} ]

故：

[tag{1}|A|^T=|A| ]

5.2 性质2：将行列式任意两行（或两列）进行互换，行列式变号

性质2的证明：

若(|A|=sum (-1)^ta_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...a_{jp_j}...a_{kp_k}...cdot a_{np_n})

设将|A|中第(j)行和第(k)行进行互换形成的行列式为(|A^{kj}|)：

相对(|A|)而言，(|A^{kj}|)中的全排列变为：(p_1p_2p_3...p_k...p_j...p_n)，即产生（或减少）了1个逆序数：

[\ Rightarrow |A^{kj}|=sum (-1)^{tpm1}a_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...a_{kp_k}...a_{jp_j}...cdot a_{np_n}\ ]

[Rightarrowtag{2} |A^{kj}|=-|A| ]

5.3 性质3：若行列式任意两行的值完全相同，则行列式的值为0

性质3的证明：

根据性质2可知：(|A^{kj}|=-|A|)

若(|A|)中(k,j)两行的值完全相同，则：

[|A^{kj}|=|A|\ Rightarrow |A| =-|A| ]

[Rightarrowtag{3} |A| = 0 ]

5.4 性质4：(lambda cdot |A|=lambda) 乘以 (|A|)中任意一行（或任意一列）的元素

性质4的证明：

由(|A|=sum (-1)^ta_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...cdot a_{np_n})可知：

[lambda cdot |A|=sum (-1)^t cdot lambda cdot a_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...cdot a_{np_n} ]

由性质1可知：(|A|=|A|^T)

[Rightarrow lambda cdot |A|=lambda cdot |A|^T=sum (-1)^ta_{p_11}cdot a_{p_22}cdot a_{p_33}cdot ... a_{p_nn} ]

又知(p_1p_2p_3......p_n) 是|A|中的全排列，故(a_{ip_i})表示第i行任一元素，(a_{p_ii})表示第i列任一元素（i=1,2,3...,n），则：

[lambda cdot |A|=sum (-1)^t cdot (lambda cdot a_{1p_1})cdot a_{2p_2}cdot a_{3p_3}...cdot a_{np_n}\ qquadquad=sum (-1)^t cdot a_{1p_1}cdot a_{2p_2}cdot a_{3p_3}...cdot (lambda cdot a_{np_n})\ ]

[qquadquadquadquad=lambda cdot |A|^T=sum (-1)^t(lambda cdot a_{p_11})cdot a_{p_22}cdot a_{p_33}cdot ... a_{p_nn}\ qquadquadquadquad......\ qquadquadquadquad=lambda cdot |A|^T=sum (-1)^ta_{p_11}cdot a_{p_22}cdot a_{p_33}cdot ... (lambda cdot a_{p_nn})\ qquadquadquadquad...... ]

[Rightarrow lambda cdot |A|= begin{vmatrix} lambda cdot a_{11} & lambda cdot a_{12} & lambda cdot a_{13} &...& lambda cdot a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ ]

[tag{4}qquadqquad= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ lambda cdot a_{n1} & lambda cdot a_{n2} & lambda cdot a_{n3} &...& lambda cdot a_{nn}\ end{vmatrix} ]

[=lambda cdot|A|^T= begin{vmatrix} lambda cdot a_{11} & a_{12} & a_{13} &...& a_{1n}\ lambda cdot a_{21} & a_{22} & a_{23} &...& a_{2n}\ lambda cdot a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ lambda cdot a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ ]

[...... ]

[tag{5} qquadqquad = begin{vmatrix} a_{11} & a_{12} & a_{13} &...& lambda cdot a_{1n}\ a_{21} & a_{22} & a_{23} &...& lambda cdot a_{2n}\ a_{31} & a_{32} & a_{33} &...& lambda cdot a_{3n}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& lambda cdot a_{nn}\ end{vmatrix} ]

5.5 性质5：若行列式任意两行（或两列）元素成比例，则行列值为0

设(|A|)存在第(x)行：

[|A|= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ ]

若(|A|)中满足：(a_{1i}=lambda cdot a_{xi}(i=1,2,3,...,n))，则(|A|)可写为如下形式：

[|A|= begin{vmatrix} lambda cdot a_{x1} & lambda cdot a_{x2} & lambda cdot a_{x3} &...& lambda cdot a_{xn}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ ]

[=lambda cdot begin{vmatrix} a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix} ]

则根据性质4，可得：

[lambda cdot begin{vmatrix} a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}=lambda cdot0=0 qquad qquad qquad(6) \ ]

5.6 性质6：行列式的两行（或两列）相加

设(|A|)存在第(z)行：

[|A|= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{z1} & a_{z2} & a_{z3} &...& a_{zn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ ]

若第z行元素均满足：(a_{z_i}=a_{x_i}+a_{y_i};(i=1,2,3,...,n))
则：

[|A|=sum(-1)^ta_{1p_1} cdot a_{2p_2} cdot ... cdot a_{zp_z}cdot ... cdot a_{np_n}\ qquad qquad quad=sum(-1)^ta_{1p_1} cdot a_{2p_2} cdot ... cdot (a_{xp_x}+a_{yp_y})cdot ... cdot a_{np_n}\ qquad qquad quad=sum(-1)^ta_{1p_1} cdot a_{2p_2} cdot ... cdot a_{xp_x}cdot ... cdot a_{np_n}+sum(-1)^ta_{1p_1} cdot a_{2p_2} cdot ... cdot a_{yp_y}cdot ... cdot a_{np_n}\ ]

则(|A|)可具有以下性质：

[|A|= tag{7} begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{z1} & a_{z2} & a_{z3} &...& a_{zn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix}\ =begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{x1} & a_{x2} & a_{x3} &...& a_{xn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix} +begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3n}\ & & ......\ a_{y1} & a_{y2} & a_{y3} &...& a_{yn}\ & & ......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nn}\ end{vmatrix} ]

同理，可通过性质1证得以下性质（过程略）：

[begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1z} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2z} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3z} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nz} & ... & a_{nn}\ end{vmatrix}\ =begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1x} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2x} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3x} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nx} & ... & a_{nn}\ end{vmatrix} +begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1y} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2y} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3y} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& a_{ny} & ... & a_{nn}\ end{vmatrix} ]

5.6.1 性质6推论：使行列式中两行（或两列）相加，但保持行列式值不变的方法：

设行列式|A|中存在第x列、第y列：

[|A|= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1x} & ... & a_{1y} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2x} & ... & a_{2y} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3x} & ... & a_{3y} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nx} & ... & a_{ny} & ... & a_{nn}\ end{vmatrix} ]

若使第x列的数值产生以下变化，则|A|的值保持不变：

[tag {8} 由a_{ix}变为(a_{ix}+lambda cdot a_{iy})quad(i=1,2,3,...,n) ]

则新生成的行列式(|A|')为：

[|A|'= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& (a_{1x}+lambda cdot a_{1y}) & ... & a_{1y} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& (a_{2x}+lambda cdot a_{2y}) & ... & a_{2y} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& (a_{3x}+lambda cdot a_{3y}) & ... & a_{3y} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& (a_{nx}+lambda cdot a_{ny}) & ... & a_{ny} & ... & a_{nn}\ end{vmatrix} ]

根据性质6可得：

[|A|'= begin{vmatrix} a_{11} & a_{12} & a_{13} &...& a_{1x} & ... & a_{1y} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& a_{2x} & ... & a_{2y} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& a_{3x} & ... & a_{3y} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& a_{nx} & ... & a_{ny} & ... & a_{nn}\ end{vmatrix}\ + begin{vmatrix} a_{11} & a_{12} & a_{13} &...& lambda cdot a_{1y} & ... & a_{1y} & ... & a_{1n}\ a_{21} & a_{22} & a_{23} &...& lambda cdot a_{2y} & ... & a_{2y} & ... & a_{2n}\ a_{31} & a_{32} & a_{33} &...& lambda cdot a_{3y} & ... & a_{3y} & ... & a_{3n}\ & & &......\ a_{n1} & a_{n2} & a_{n3} &...& lambda cdot a_{ny} & ... & a_{ny} & ... & a_{nn}\ end{vmatrix} ]

[Rightarrow |A|'=|A|+0=|A| ]

同理，设行列式|A|中存在第x行、第y行，若使x行产生以下变化，则|A|的值保持不变：

[tag{9} 由a_{xj}变为(a_{xi}+lambda cdot a_{yi})quad(i=1,2,3,...,n) ]

5.7 性质7：(|A|cdot |B|=|Acdot B|)

（证明过程参考后续知识点）