行列式和它的转置行列式值相等，即：

∣ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ = ∣ a 11 a 21 . . . a n 1 a 12 a 22 . . . a n 2 . . . . . . . . . . . . a 1 n a 2 n . . . a n n ∣ \left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| = \left|\begin{array}{ccc} a_{11} & a_{21} & ... & a_{n1} \\ a_{12} & a_{22} & ... & a_{n2} \\ . & . & ... & . \\ . & . & ... & . \\ a_{1n} & a_{2n} & ... & a_{nn} \end{array} \right| ​a11​a21​..an1​​a12​a22​..an2​​...............​a1n​a2n​..ann​​ ​= ​a11​a12​..a1n​​a21​a22​..a2n​​...............​an1​an2​..ann​​ ​

互换行列式的任意两行（两列），行列式的值将改变正负号：

∣ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ = − ∣ a 21 a 22 . . . a 2 n a 11 a 12 . . . a 1 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ \left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| = -\left|\begin{array}{ccc} a_{21} & a_{22} & ... & a_{2n} \\ a_{11} & a_{12} & ... & a_{1n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| ​a11​a21​..an1​​a12​a22​..an2​​...............​a1n​a2n​..ann​​ ​=− ​a21​a11​..an1​​a22​a12​..an2​​...............​a2n​a1n​..ann​​ ​

行列式某行或者某列的公因子可以提到行列式记号外面：

∣ b 1 a 11 b 1 a 12 . . . b 1 a 1 n b 2 a 21 b 2 a 22 . . . b 2 a 2 n . . . . . . . . . . . . b n a n 1 b n a n 2 . . . b n a n n ∣ = ∏ i = 1 n b i ∣ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ \left|\begin{array}{ccc} b_1a_{11} & b_1a_{12} & ... & b_1a_{1n} \\ b_2a_{21} & b_2a_{22} & ... & b_2a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ b_na_{n1} & b_na_{n2} & ... & b_na_{nn} \end{array} \right| = \prod_{i=1}^n b_i\left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| ​b1​a11​b2​a21​..bn​an1​​b1​a12​b2​a22​..bn​an2​​...............​b1​a1n​b2​a2n​..bn​ann​​ ​=∏i=1n​bi​ ​a11​a21​..an1​​a12​a22​..an2​​...............​a1n​a2n​..ann​​ ​

行列式具有分行（列）相加性：

∣ b 1 + c 1 b 2 + c 2 . . . b n + c n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ = ∣ b 1 b 2 . . . b n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ + ∣ c 1 c 2 . . . c n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ \left|\begin{array}{ccc} b_1+c_1 & b_2+c_2 & ... & b_n+c_n \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| = \left|\begin{array}{ccc} b_1 & b_2 & ... & b_n \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| + \left|\begin{array}{ccc} c_1 & c_2 & ... & c_n \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| ​b1​+c1​a21​..an1​​b2​+c2​a22​..an2​​...............​bn​+cn​a2n​..ann​​ ​= ​b1​a21​..an1​​b2​a22​..an2​​...............​bn​a2n​..ann​​ ​+ ​c1​a21​..an1​​c2​a22​..an2​​...............​cn​a2n​..ann​​ ​

将行列式某一行（列）的各元素同乘以一个数 k k k后加到另外一行（列）其值不变。这一点同矩阵初等变换

∣ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ = ∣ a 11 a 12 . . . a 1 n a 21 + k a 11 a 22 + k a 12 . . . a 2 n + k a 1 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ \left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| = \left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} + ka_{11} & a_{22}+ka_{12} & ... & a_{2n}+ka_{1n} \\ . & . & ... & . \\ . & . & ... & . \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| ​a11​a21​..an1​​a12​a22​..an2​​...............​a1n​a2n​..ann​​ ​= ​a11​a21​+ka11​..an1​​a12​a22​+ka12​..an2​​...............​a1n​a2n​+ka1n​..ann​​ ​

拉普拉斯定理：行列式等于它的任一行（列）的各元素与其对应的代数余子式的乘积之和：

∣ A ∣ = ∑ i = 1 n ( − 1 ) 1 + i a 1 i C 1 i |A| = \sum_{i=1}^n(-1)^{1+i}a_{1i}C_{1i} ∣A∣=∑i=1n​(−1)1+ia1i​C1i​

分块行列式的值等于其主对角线上两个子块行列式的值的乘积（右上角的块必须为全零）：

∣ a 11 a 12 0 0 a 21 a 22 0 0 b 11 b 12 c 11 c 12 b 21 b 22 c 21 c 22 ∣ = ∣ a 11 a 12 a 21 a 22 ∣ ∣ c 11 c 12 c 21 c 22 ∣ \left|\begin{array}{ccc} a_{11} & a_{12} & 0 & 0 \\ a_{21} & a_{22} & 0 & 0 \\ b_{11} & b_{12} & c_{11} & c_{12} \\ b_{21} & b_{22} & c_{21} & c_{22} \end{array} \right| = \left|\begin{array}{ccc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right| \left|\begin{array}{ccc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right| ​a11​a21​b11​b21​​a12​a22​b12​b22​​00c11​c21​​00c12​c22​​ ​= ​a11​a21​​a12​a22​​ ​ ​c11​c21​​c12​c22​​ ​

上三角行列式、下三角行列式以及主对角线行列式的值都是主对角线上元素乘积。

∣ a 11 0 . . . 0 a 21 a 22 . . . 0 . . . . . 0 . . . . . 0 a n 1 a n 2 . . . a n n ∣ = ∏ i = 1 n a i i \left|\begin{array}{ccc} a_{11} & 0 & ... & 0 \\ a_{21} & a_{22} & ... & 0 \\ . & . & ... & 0 \\ . & . & ... & 0 \\ a_{n1} & a_{n2} & ... & a_{nn} \end{array} \right| = \prod_{i=1}^na_{ii} ​a11​a21​..an1​​0a22​..an2​​...............​0000ann​​ ​=∏i=1n​aii​

∣ a 11 a 12 . . . a 1 n 0 a 22 . . . a 2 n . . . . . a 3 n . . . . . . 0 0 . . . a n n ∣ = ∏ i = 1 n a i i \left|\begin{array}{ccc} a_{11} & a_{12} & ... & a_{1n} \\ 0 & a_{22} & ... & a_{2n} \\ . & . & ... & a_{3n} \\ . & . & ... & . \\ 0 & 0 & ... & a_{nn} \end{array} \right|= \prod_{i=1}^na_{ii} ​a11​0..0​a12​a22​..0​...............​a1n​a2n​a3n​.ann​​ ​=∏i=1n​aii​

∣ a 11 0 . . . 0 0 a 22 . . . 0 . . . . . 0 . . . . . . 0 0 . . . a n n ∣ = ∏ i = 1 n a i i \left|\begin{array}{ccc} a_{11} & 0 & ... & 0 \\ 0 & a_{22} & ... & 0 \\ . & . & ... & 0 \\ . & . & ... & . \\ 0 & 0 & ... & a_{nn} \end{array} \right| = \prod_{i=1}^na_{ii} ​a11​0..0​0a22​..0​...............​000.ann​​ ​=∏i=1n​aii​

副对角线行列式的值：

∣ 0 0 . . . a 11 0 0 . . . 0 . . . . . . 0 a n − 1 , n − 1 . . . . a n n 0 . . . 0 ∣ = ( − 1 ) n ( n − 1 ) 2 ∗ ∏ i = 1 n a i i \left|\begin{array}{ccc} 0 & 0 & ... & a_{11} \\ 0 & 0 & ... & 0 \\ . & . & ... & . \\ 0 & a_{n-1,n-1} & ... & . \\ a_{nn} & 0 & ... & 0 \end{array} \right| = (-1)^{\frac{n(n-1)}{2}} *\prod_{i=1}^na_{ii} ​00.0ann​​00.an−1,n−1​0​...............​a11​0..0​ ​=(−1)2n(n−1)​∗∏i=1n​aii​

若行列式有两行（列）元素对应相等，那么此行列式的值为零；若行列式有一行（列）元素全为零，那么此行列式的值为零若行列式有两行（列）元素成比例，那么此行列式的值为零。

定义法

计算较为复杂一般只用于低阶行列式。

三角化法

将行列式化为上三角（下三角）行列式。

箭形行列式

形如 ∣ a a . . . a a b . . . 0 . . . . . . a 0 . . . 0 a 0 . . . x ∣ \left|\begin{array}{ccc} a & a & ... & a \\ a & b & ... & 0 \\ . & . & ... & . \\ a & 0 & ... & 0 \\ a & 0 & ... & x \end{array} \right| ​aa.aa​ab.00​...............​a0.0x​ ​的行列式，只需按主对角线上的值 t t t使得第一列减去后面每列的 1 t \frac{1}{t} t1​。

降阶法

根据拉普拉斯展开定理，对行列式进行适当的展开。

升阶法

将原行列式增加一行一列，而保持原行列式的值不变或与原行列式有某种巧妙的关系。

例：

∣ x a . . . a a x . . . a . . . . . . a a . . . a a a . . . x ∣ = ∣ 1 a a . . . a 0 x a . . . a 0 a x . . . a 0 . . . . . . 0 a a . . . a 0 a a . . . x ∣ = ∣ 1 a a . . . a − 1 x − a 0 . . . 0 − 1 0 x − a . . . 0 − 1 . . . . . . − 1 0 0 . . . 0 − 1 0 0 . . . x − a ∣ \left|\begin{array}{ccc} x & a & ... & a \\ a & x & ... & a \\ . & . & ... & . \\ a & a & ... & a \\ a & a & ... & x \end{array} \right| = \left|\begin{array}{ccc} 1 & a& a &...& a\\ 0 & x & a & ... & a \\ 0 & a & x & ... & a \\ 0 & . & . & ... & . \\ 0 & a & a & ... & a \\0 & a & a & ... & x \end{array} \right| = \left|\begin{array}{ccc} 1 & a& a &...& a\\ -1 & x-a & 0 & ... & 0 \\ -1 & 0 & x-a & ... & 0 \\ -1 & . & . & ... & . \\ -1 & 0 & 0 & ... & 0 \\-1 & 0 & 0 & ... & x-a \end{array} \right| ​xa.aa​ax.aa​...............​aa.ax​ ​= ​100000​axa.aa​aax.aa​..................​aaa.ax​ ​= ​1−1−1−1−1−1​ax−a0.00​a0x−a.00​..................​a00.0x−a​ ​

然后 C 1 + 1 x − a C 2 , . . . c 1 + 1 x − a C n C_1+\frac{1}{x-a}C_2,...c_1+\frac{1}{x-a}C_n C1​+x−a1​C2​,...c1​+x−a1​Cn​，最后化简为： ∣ 1 + n a x − a a a . . . a 0 x − a 0 . . . 0 0 0 x − a . . . 0 0 . . . . . . 0 0 0 . . . 0 0 0 0 . . . x − a ∣ \left|\begin{array}{ccc} 1+\frac{na}{x-a} & a& a &...& a\\ 0 & x-a & 0 & ... & 0 \\ 0 & 0 & x-a & ... & 0 \\ 0 & . & . & ... & . \\ 0 & 0 & 0 & ... & 0 \\0 & 0 & 0 & ... & x-a \end{array} \right| ​1+x−ana​00000​ax−a0.00​a0x−a.00​..................​a00.0x−a​ ​

递推法

找出行列式和其相应的 n − 1 , n − 2 , . . . n-1,n-2,... n−1,n−2,...阶行列式之间的递推关系。

数学归纳法

由行列式的特殊形式，计算低阶行列式的公式猜想推广到高阶行列式。

拆分法

根据行列式的某些位置由若干数相加，那么根据行列式的分行（列）相加性拆分成多个行列式求解。

分解乘积法

根据行列式的特点利用行列式的乘法公式把所给行列式分解成两个易求解的行列式之积，通过对这两个行列式的计算从而得到其值。

∣ a 1 + b 1 a 1 + b 2 . . . a 1 + b n a 2 + b 1 a 2 + b 2 . . . a 2 + b n . . . . . . . . . . . . a n + b 1 a n + b 2 . . . a n + b n ∣ = ∣ a 1 1 0 . . . 0 a 2 1 0 . . . 0 . . . . . . . . . . . . . . a n 1 0 . . . 0 ∣ ∗ ∣ b 1 b 2 b 3 . . . b n 1 1 1 . . . 1 0 0 0 . . . 0 . . . . . . . 0 0 0 . . . 0 ∣ = { a 1 + b 1 n = 1 ( a 1 − a 2 ) ( b 2 − b 1 ) n = 2 0 n ≥ 3 \left|\begin{array}{ccc} a_1+b_1 & a_1+b_2 & ... & a_1+b_n \\ a_2+b_1 & a_2+b_2 & ... & a_2+b_n \\ . & . & ... & . \\ . & . & ... & . \\ a_n+b_1 & a_n+b_2 & ... & a_n+b_n \end{array} \right| = \left|\begin{array}{ccc} a_1 & 1 & 0 &... & 0 \\ a_2 & 1 & 0 & ... & 0 \\ . & . & . & ... & . \\ . & . & . & ... & . \\ a_n & 1 & 0 & ... & 0 \end{array} \right| * \left|\begin{array}{ccc} b_1 & b_2 & b_3 &... & b_n \\ 1 & 1 & 1 & ... & 1 \\ 0 & 0 & 0 & ... & 0 \\ . & . & . & ... & . \\ 0 & 0 & 0 & ... & 0 \end{array} \right|= \left\{\begin{array}{rcl} a_1+b_1 & n =1\\ (a_1-a_2)(b_2-b_1) & n=2\\ 0 & n\geq3 \end{array}\right. ​a1​+b1​a2​+b1​..an​+b1​​a1​+b2​a2​+b2​..an​+b2​​...............​a1​+bn​a2​+bn​..an​+bn​​ ​= ​a1​a2​..an​​11..1​00..0​...............​00..0​ ​∗ ​b1​10.0​b2​10.0​b3​10.0​...............​bn​10.0​ ​=⎩ ⎨ ⎧​a1​+b1​(a1​−a2​)(b2​−b1​)0​n=1n=2n≥3​