# How to calculate these matrices? - explanation of the procedure

$A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 2 \\ 3 & 0 & 1 \end{array}\right)$, $B=\left(\begin{array}{ccc} 2 & 1 \\ 0 & 1 \\ 3 & 0 \end{array}\right)$ and $C=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 2 \end{array}\right)$.

## The task is to calculate these matrices: $(CA^{-1})$ and $(A^{-1}B)$.

In my textbook there is a hint. According to the author we should calculate $(A^T|C^T)$ to gain $X^{T}=(CA^{-1})^{T}$. Then it should be easy to convert back to $(CA^{-1})$.

Regarding $(A^{-1}B)$ there is a hint: "Use $(A|B)$."

I tried to use these hints a while ago and both worked well. The problem is that I do not know why I used it. Can you please explain to me this procedure?

Thanks

When you go from $(A|B)$ to $(I|D)$ by doing elementary row operations (which you didn't say, but I assume that's what you mean), each elementary row operation corresponds to multiplying both $A$ and $B$ by some elementary matrix $E_i$. On the left, you wind up with $E_nE_{n-1}\dots E_2E_1A$, but you also wind up with $I$, so $E_nE_{n-1}\dots E_2E_1=A^{-1}$. So $D=E_nE_{n-1}\dots E_2E_1B=A^{-1}B$.
Here $A$ is an elementary matrix so applying $A^{-1}$ from right to any other matrix would mean that you are applying the inverse operations (inverse to what was done in case of columns when applying A itself to that matrix from the right ) on the columns of that matrix.