二阶矩阵求逆的口诀及其应用 矩阵求逆有很多应用 , 是高等代数中的重要内容 , 通常有两个方法 : 伴随矩阵法与 初等变换法 . 例 1. 求 A = \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d 5 4 0 3 2 0 0 0 3 的逆矩阵 . 解一 ( 伴随矩阵法 ) 先求 A 的行列式 : | A |= 5 4 0 3 2 0 0 0 3 \uf02d \uf02d =3 5 4 3 2 \uf02d \uf02d =3(-10+12)=6 ≠ 0. 再求 A 的代数余子式 : A 11 = 5 4 3 2 \uf02d \uf02d =2, A 5 0 3 0 12 \uf02d \uf02d \uf02d \uf03d =0, A 4 0 2 0 13 \uf03d =0, A 5 4 0 0 21 \uf02d \uf02d \uf03d =0, A 5 0 0 3 22 \uf02d \uf03d =-15, A 4 0 0 3 23 \uf02d \uf03d =-12, A 3 2 0 0 31 \uf02d \uf03d =0, A 3 0 0 3 32 \uf02d \uf02d \uf03d =9, A 2 0 0 3 33 \uf03d =6. 于是可求得 A \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf03d \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf03d \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf03d \uf02d 1 2 0 0 0 0 6 12 0 9 15 0 0 0 2 6 1 1 2 3 2 5 3 1 33 23 13 32 22 12 23 21 11 1 A A A A A A A A A A . 解二 ( 初等变换法 ) 将 \uf028 \uf029 AE 初等变换为 \uf028 \uf029 1 \uf02d EA , 即可求得 A 的逆 : \uf028 \uf029 \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf0ae \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf0ae \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf0ae \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf03d 1 2 0 0 0 0 1 0 0 0 1 0 0 0 1 1 2 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 5 4 0 3 2 0 0 0 1 1 0 0 0 1 0 0 0 1 5 4 0 3 2 0 0 0 3 2 3 2 5 3 1 2 1 3 1 2 3 3 1 AE ∴ A \uf0f7 \uf0f7 \uf0f7 \uf0f8 \uf0f6 \uf0e7 \uf0e7 \uf0e7 \uf0e8 \uf0e6 \uf02d \uf02d \uf03d \uf02d 1 2 0 0 0 0 2 3 2 5 3 1 1 . 显然 , 这两种