Question

用逆矩阵解下列矩阵方程。逆矩阵

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Answer 1

第1题

第2题

Answer 2

(1) XA = B, X = BA^(-1)
(A, E) =
[1 1 1 1 0 0]
[0 1 1 0 1 0]
[0 0 1 0 0 1]
初等行变换为
[1 1 0 1 0 -1]
[0 1 0 0 1 -1]
[0 0 1 0 0 1]
初等行变换为
[1 0 0 1 -1 -1]
[0 1 0 0 1 -1]
[0 0 1 0 0 1]
A^(-1) =
[1 -1 -1]
[0 1 -1]
[0 0 1]
X = BA^(-1) =
[1 -3 2]
[0 1 -2]
(2) AXB = C, X = A^(-1)CB^(-1)
B^(-1) =
[ 3 -1]
[-5 2]
(A, E) =
[1 2 3 1 0 0]
[2 2 1 0 1 0]
[3 4 3 0 0 1]
初等行变换为
[1 2 3 1 0 0]
[0 -2 -5 -2 1 0]
[0 -2 -6 -3 0 1]
初等行变换为
[1 0 -2 -1 1 0]
[0 -2 -5 -2 1 0]
[0 0 -1 -1 -1 1]
初等行变换为

[1 0 0 1 3 -2]
[0 -2 0 3 6 -5]
[0 0 1 1 1 -1]
初等行变换为
[1 0 0 1 3 -2]
[0 1 0 -3/2 -3 5/2]
[0 0 1 1 1 -1]
A^(-1) =
[ 1 3 -2]
[-3/2 -3 5/2]
[ 1 1 -1]
A^(-1)C =
[1 1]
[0 -2]
[0 4]
X = A^(-1)CB^(-1) =
[ -2 1]
[ 10 -4]
[-20 8]