SOLUTION: Assume that all matrices are invertible and solve for the matrix X in: (A^-1 X)^-1 =A (B^-1 A)^-1

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Question 624750: Assume that all matrices are invertible and solve for the matrix X in:
(A^-1 X)^-1 =A (B^-1 A)^-1
Answer by jim_thompson5910(28536) About Me  (Show Source):
You can put this solution on YOUR website!
(A^-1 X)^-1 =A (B^-1 A)^-1

(A^-1 X)^-1(A^-1 X) =(A (B^-1 A)^-1)(A^-1 X)

I =(A (B^-1 A)^-1)(A^-1 X)

A^-1 * I =(A^-1 A (B^-1 A)^-1)(A^-1 X)

A^-1 = ((B^-1 A)^-1)(A^-1 X)

(B^-1 A)A^-1 = ((B^-1 A)(B^-1 A)^-1)(A^-1 X)

(B^-1 A)A^-1 = A^-1 X

A(B^-1 A)A^-1 = A*A^-1 X

A(B^-1 A)A^-1 = X

X = A(B^-1 A)A^-1

So the solution is X = A(B^-1 A)A^-1