### Instruction: Find the matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = -5 \mathbf{v}\]for all vectors $\mathbf{v}.$.### Response: I know that this means $\mathbf{M}$ is a scalar multiple of the identity matrix $\mathbf{I},$ since $\mathbf{I} \mathbf{v} = \mathbf{v}$ for all vectors $\mathbf{v}.$
So I can write $\mathbf{M} = k \mathbf{I}$ for some scalar $k.$
To find $k,$ I can plug in any vector $\mathbf{v}$ and compare the results.
For example, let $\mathbf{v} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}.$
Then $\mathbf{M} \mathbf{v} = k \mathbf{I} \mathbf{v} = k \mathbf{v} = -5 \mathbf{v}.$
So $k = -5.$
Therefore, $\mathbf{M} = -5 \mathbf{I} = -5 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -5 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -5 \end{bmatrix}.$