Fp1 matrix algebra inverses

Hey, I am a bit confused here. Hoping someone with a knowledge of matrices can help me out.

As matrix algebra is not commutative, how do I know which side to put the A^-1 on of the matrices to ensure I get the right answer when solving these sort of problems? (shown in the picture)

I hope I have conveyed myself clearly

With matrix equations, you can either multiply the equation by a matrix either 'on the left' or 'on the right'.


E.g. $AC = B$


If you multiply both sides of the equation by $A^{-1}$ on the left, you get

$A^{-1} A C = A^{-1} B$

and the $A^{-1}$ and $A$ cancel to give

$C = A^{-1} B$


If instead you were to multiply the original equation by $A^{-1}$ on the right, then you would get

$ACA^{-1} = BA^{-1}$

but this is not useful here since no matrices cancel.


So for matrix equations, you always need to choose the best direction to multiply your equation by. Does that help?