Question

4. Given that A and B are invertible matrices of the same size and that A(2X + B)? A-1 = (AB-1A-1)-1, solve the equation for X(you have to simplify the answer).

Answer 1

4. Since (PQ)^-1 = Q^-1 P^-1, we have (AB^-1 A^-1)^-1 = (A^-1)^-1(AB^-1)^-1 = A (B^-1)^-1 A^-1 = ABA^-1.

Thus, the givenm equation changes to A(2X+B)^T A^-1 = ABA^-1.

Now, on multiplying both the sides to the left by A^-1 to to the right by A, we get (2X+B)^T = B so that 2X+B = B^T or, 2X = B^T -B. Hence X = (1/2)(B^T -B).