Question

If a, b, and c are nxn invertible matrices, does the equation c^-1(a+x)b^-1 have a solution c

Answer 1

There is a solution. Starting from C−1(A+X)B−1=In C−1(A+X)B−1=In, multiply both of the sides of the equation, by C and multiply both sides of the equation by B.

In other terms, this is the solution:

Given: C^(-1)(A + X)B^(-1) = In

= CC^(-1)(A + X)B^(-1)B = CInB

= In(A + X)In = CB

= AInIn + XInIn = CB


= A + X = CB


X = CB – A

 

The final answer Is X = CB – A