I'm solving some exercises that I found online relating to transformation matrices and change of basis matrices. I'm having trouble understanding the last part of the answer.
Here's the question:
Let $A = \begin{bmatrix}5&-3\\2&-2\end{bmatrix}$, which is a linear transformation $\mathbb{R}^2 \rightarrow \mathbb{R}^2$. Find the matrix representing the transformation with respect to basis B = $\begin{bmatrix}3&1\\1&2\end{bmatrix}$.
I understand how to solve the question. You would do $B^{-1}AB$. I did the multiplication and got $\begin{bmatrix}20&0\\0&-5\end{bmatrix}$. The author of the document got the same answer but then multiplied the matrix by 1/5 to get $\begin{bmatrix}4&0\\0&-1\end{bmatrix}$. Why is he allowed to do that?
You made a mistake: $1/5$ is included into $B^{-1}$, which you forgot.