Inverse Matrix Multiplication Problem

Question

Problem:

In this problem, I multiplied the left side of the AX=B equation by the inverse of A, but what I am confused about is why then I must multiply B by the inverse of A in this order: (inverse of A) * (B). When I do (B) * (inverse of A), it is wrong, but why?

Answer 1

We have that

$$AX=B$$

and by left multiplication by $A^{-1}$ and associativity we obtain

$$A^{-1}(AX)=A^{-1}B\implies (A^{-1}A)X=A^{-1}B\implies X=A^{-1}B$$

Recall also that matrix multiplication is not commutative and moreover in that case the multiplication $BA^{-1}$ is not well defined since the dimensions of the two matrices, for a multiplication in that order, are not compatible.

Answer 2

There's no need to multiply by anything at all. An $m\times n$ matrix multiplied by an $n\times k$ matrix will be an $m\times k$ matrix. We cannot multiply an $m\times n$ matrix by a $p\times k$ matrix unless $n=p.$ Use this to find the size of $X,$ instead.