What values of $k$ if any will make matrices $AB = BA$?

Question

$$A = \begin{bmatrix} 2 & 1\\ 1 & -1\\ \end{bmatrix}$$

and

$$B = \begin{bmatrix} 1 & k\\ -1 & 2\\ \end{bmatrix}$$

I need to find a value of $k$ that will make $AB = BA$.

I multiplied the two matrices and found no such value of k. Does that mean that there is no value of k that makes $AB = BA$?

Answer 1

You are right.

No matter what $k$ is, the lower left element of $AB$ will be $2$ and the same element of $BA$ will be $0$. So these are never equal.