Sum of two inverse matrices

Question

I having trouble working this simple problem out:

Given 3 inverse matrices:

$A = \begin{bmatrix}2&&2\\3&&4\end{bmatrix}, B =\begin{bmatrix}1&&2\\-1&&2\end{bmatrix} \& \quad C = \begin{bmatrix}4&&-2\\1&&0\end{bmatrix} $

Calculate the solution of $AX + BX = C$ , algebraically?

I know the answer is $AX + BX = C \iff (A+B)^{-1}\cdot C$

Can someone help me out with this? I can't figure out the steps taken.

Answer 1

I'm not sure if this is what you're after, but I'll give it a try:\begin{align}AX+BX=C&\iff(A+B)X=C\\&\iff(A+B)^{-1}\bigl((A+B)X\bigr)=(A+B)^{-1}C\\&\iff\bigl((A+B)^{-1}(A+B)\bigr)X=(A+B)^{-1}C\\&\iff\operatorname{Id}.X=(A+B)^{-1}C\\&\iff X=(A+B)^{-1}C.\end{align}

Answer 2

$$AX+BX = C \\ (A+B)X = C \\ (A+B)^{-1}(A+B)X = (A+B)^{-1}C \\ X = (A+B)^{-1}C$$