I'm working on a problem and I'm a bit confused because I know order matters in matrix multiplication. Consider the following question.
Let A be a (2x2) matrix and let $$ B=\begin{bmatrix} 1 & 3 \\ 1 & 4 \end{bmatrix} and\ C=\begin{bmatrix} 2 & 3\\ 4 & 5 \end{bmatrix} $$
If $A^{T}$B=C, what is A?
I assume we multiply both sides by $B^{-1}$ but in this case would the right side of the expression equal $B^{-1}$C or C$B^{-1}$?
You want to have $BB^{-1}$ on the left hand side to produce $$A^{T}BB^{-1}$$ so you multiply the $B^{-1}$ to the right on the RHS as well $$A^{T}BB^{-1} = CB^{-1}$$