If A B and C are n x n invertible matrices does the equation

If A, B, and C are n x n invertible matrices, does the equation C (AB + CX) = I_n have a solution X? If so, find it. If not, explain your reasoning, (note that X is a MATRIX; also, pay attention to left hand vs. right hand multiplication of matrices.)

Solution

C^-1(AB+CX)=I_n

CC^-1(AB+CX)=C I_n (Multiply both side by C matrix)

(AB+CX)=C ( CC^-1=I_n & C I_n=C )

CX=C-AB

C^-1CX=C^-1(C-AB) ( Multiply both side by C^-1 matrix)

X= C^-1(C-AB) (C^-1C =I_n & I_n X=X)