# Does multiplying a transformation matrix by a scalar change the transformation?

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.