So this question is about multiplying both sides of a matrix equation by a matrix.
Let's say I have this equation: $M(AB + CD) = X(EF + GH)$
and I want to make D free from any multiplication, which method is correct?
$M(A C^{-1}B + D) = X(EC^{-1}F + GC^{-1}H) $
$M(C^{-1}AB + D) = X(C^{-1}EF + C^{-1}GH) $
$C^{-1}M(AB + CD) = C^{-1} X(EF + GH) $ (I premultiplied both sides with $C^{-1}$ but I don't know how that will remove the $C$ from the $D$)
Also, where can I find more resources about multiplying equations with the inverse of matrix to get rid of it? That is easy for normal algebra, but I think it is more complicated for matrices.
Assumptions:
-All matrices are invertible.
-All matrices are of the same size ($n $ by $ n$)
Thanks
Best Answer
Expand $$MAB + MCD = XEF+XGH$$ Subtract $MAB$ $$MCD = XEF+XGH-MAB$$ Multiply by $C^{-1}M^{-1}$ on the right $$D=C^{-1}M^{-1}(XEF+XGH-MAB)$$