Question

线性代数,矩阵,对的肯定采纳的

Answer 1

先以四阶矩阵为例,(A,E) =

[1 1 1 1 1 0 0 0]

[0 1 1 1 0 1 0 0]

[0 0 1 1 0 0 1 0]

[0 0 0 1 0 0 0 1]

行初等变换为

[1 1 1 0 1 0 0 -1]

[0 1 1 0 0 1 0 -1]

[0 0 1 0 0 0 1 -1]

[0 0 0 1 0 0 0 1]

行初等变换为

[1 1 0 0 1 0 -1 0]

[0 1 0 0 0 1 -1 0]

[0 0 1 0 0 0 1 -1]

[0 0 0 1 0 0 0 1]

行初等变换为

[1 0 0 0 1 -1 0 0]

[0 1 0 0 0 1 -1 0]

[0 0 1 0 0 0 1 -1]

[0 0 0 1 0 0 0 1]

则对于 n 阶矩阵,A^(-1) =

[1 -1 0 ...0]

[0 1 -1 ...0]

[.]

[0 0 0 ...-1]

[0 0 0 ...1]