Question

Let A,B, and C be square invertible matrices of the same size. If C has no eigenvalue equal to -1, then (AB + ACB)^-1 is equal to:
a. B^-1 (I + C)^-1*A^-1
b. (I + C)^-1*B^-1*A^-1
c. A^-1B*(I + C)^-1
d. (I + C)^-1*BA^-1

Answer 1

AB + ACB = A (B + CB) = A (I + C ) B

Taking the inverse gives

(A (I + C ) B)⁻¹ = B ⁻¹ (I + C )⁻¹ A ⁻¹

so the answer is (A)