Question

应用逆矩阵解下列矩阵方程，要过程，非常感谢

Answer 1

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

(2) PXQ = A, X = P^(-1)AQ^(-1)
P, Q 都是交换变换的初等矩阵，则
P^(-1) = P, Q^(-1) = Q
X = P^(-1)AQ^(-1) = PAQ =
[2 -1 0]
[1 3 4]
[0 0 -2]