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

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