Question

1.用初等变换化下列矩阵成最简阶梯矩阵 1 1 2 1 2 -1 2 4 1 -2 0 3 4 1 4 2

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2.用初等变换求下列矩阵的逆矩阵
1 0 1
3 1 0
-3 2 5

Answer 1

1. A=
[1 1 2 1]
[2 -1 2 4]
[1 -2 0 3]
[4 1 4 2]
行初等变换为
[1 1 2 1]
[0 -3 -2 2]
[0 -3 -2 2]
[0 -3 -4 -2]
行初等变换为
[1 1 2 1]
[0 3 2 -2]
[0 0 -2 -4]
[0 0 0 0]
行初等变换为
[1 1 0 -3]
[0 3 0 -6]
[0 0 1 2]
[0 0 0 0]
行初等变换为
[1 0 0 -1]
[0 1 0 -2]
[0 0 1 2]
[0 0 0 0]
2. (A,E)=
[1 0 1 1 0 0]
[3 1 0 0 1 0]
[-3 2 5 0 0 1]
行初等变换为
[1 0 1 1 0 0]
[0 1 -3 -3 1 0]
[0 2 8 3 0 1]
行初等变换为
[1 0 1 1 0 0]
[0 1 -3 -3 1 0]
[0 0 14 9 -2 1]
行初等变换为
[1 0 0 5/14 1/7 -1/14]
[0 1 0 -15/14 4/7 3/14]
[0 0 1 9/14 -1/7 1/14]
则 A^(-1)=
[ 5/14 1/7 -1/14]
[-15/14 4/7 3/14]
[ 9/14 -1/7 1/14]