第一章 行列式
设A、B为n阶矩阵
∣ A T ∣ = ∣ A ∣ \left | A^T \right | =\left | A \right | AT =∣A∣
∣ A m ∣ = ∣ A ∣ m \left | A^m \right | =\left | A \right | ^m ∣Am∣=∣A∣m
∣ k A ∣ = k n ∣ A ∣ \left | kA \right | =k^n\left | A \right | ∣kA∣=kn∣A∣
∣ A B ∣ = ∣ A ∣ ∣ B ∣ \left | AB \right | =\left | A \right | \left | B \right | ∣AB∣=∣A∣∣B∣
若 A 可逆，则 ∣ A − 1 ∣ = 1 ∣ A ∣ 若A可逆，则\left | A^{-1} \right | =\frac{1}{\left | A\right | } 若A可逆，则 A−1 =∣A∣1
∣ A ∗ ∣ = ∣ A ∣ n − 1 \left | A^* \right | =\left | A \right | ^{n-1} ∣A∗∣=∣A∣n−1
A A ∗ = A ∗ A = ∣ A ∣ E AA^*=A^*A=\left | A \right | E AA∗=A∗A=∣A∣E
A ∗ = ∣ A ∣ A − 1 ( 若 A 可逆 ) A^*=\left | A \right | A^{-1}(若A可逆) A∗=∣A∣A−1(若A可逆)
A = ∣ A ∣ ( A ∗ ) − 1 A=\left | A \right | (A^*)^{-1} A=∣A∣(A∗)−1
∣ A 1 A 2 A 3 ∣ = A 1 A 2 A 3 , ∣ A 1 A 2 A 3 ∣ = − A 1 A 2 A 3 \begin{vmatrix}A_1 & & \\ & A_2 & \\ & &A_3 \end{vmatrix}=A_1A_2A_3, \begin{vmatrix} & &A_1 \\ & A_2 & \\A_3 & & \end{vmatrix}=-A_1A_2A_3 A1A2A3 =A1A2A3, A3A2A1 =−A1A2A3
设A为n阶矩阵，B为m阶矩阵，根据拉普拉斯展开定理有
∣ A 0 0 B ∣ = ∣ A C 0 B ∣ = ∣ A 0 C B ∣ = ∣ A ∣ ∣ B ∣ \begin{vmatrix}A & 0\\0 &B \end{vmatrix}=\begin{vmatrix}A & C\\0 &B \end{vmatrix}=\begin{vmatrix}A & 0\\C &B \end{vmatrix}=\left | A \right | \left | B \right | A00B = A0CB = AC0B =∣A∣∣B∣
∣ 0 A B 0 ∣ = ∣ C A B 0 ∣ = ∣ 0 A B C ∣ = ( − 1 ) m n ∣ A ∣ ∣ B ∣ \begin{vmatrix}0 & A\\B &0 \end{vmatrix}=\begin{vmatrix}C & A\\B &0 \end{vmatrix}=\begin{vmatrix}0 & A\\B &C \end{vmatrix}=(-1)^{mn}\left | A \right | \left | B \right | 0BA0 = CBA0 = 0BAC =(−1)mn∣A∣∣B∣
第二章 矩阵
矩阵转置的性质
( A T ) T = A (A^T)^T=A (AT)T=A
( k A ) T = k A T (kA)^T=kA^T (kA)T=kAT
( A ± B ) T = A T ± B T (A\pm B)^T=A^T\pm B^T (A±B)T=AT±BT
( A B ) T = B T A T (AB)^T=B^TA^T (AB)T=BTAT
( A − 1 ) T = ( A T ) − 1 (A^{-1})^T=(A^T)^{-1} (A−1)T=(AT)−1
( A T ) m = ( A m ) T (A^T)^m=(A^m)^T (AT)m=(Am)T
矩阵伴随的性质
A ∗ = ∣ A ∣ A − 1 ( 若 A 可逆 ) A^*=\left | A \right | A^{-1}(若A可逆) A∗=∣A∣A−1(若A可逆)
A − 1 = 1 ∣ A ∣ A ∗ A^{-1}=\frac{1}{\left | A \right | } A^* A−1=∣A∣1A∗
( A T ) ∗ = ( A ∗ ) T (A^T)^*=(A^*)^T (AT)∗=(A∗)T
( k A ) ∗ = k n − 1 A ∗ (kA)^*=k^{n-1}A^* (kA)∗=kn−1A∗
( A B ) ∗ = B ∗ A ∗ (AB)^*=B^*A^* (AB)∗=B∗A∗
( a b c d ) ∗ = ( d − b − c a ) \begin{pmatrix} a & b\\ c&d \end{pmatrix}^*=\begin{pmatrix} d & -b\\ -c&a \end{pmatrix} (acbd)∗=(d−c−ba)