### Instruction: Find the $3 \times 3$ matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = -4 \mathbf{v}\]for all three-dimensional vectors $\mathbf{v}.$.### Response: I want to find a matrix $\mathbf{M}$ that multiplies any vector by $-4.$
That means that each entry of $\mathbf{M}$ must be $-4$ times the corresponding entry of the identity matrix $\mathbf{I}.$
The identity matrix $\mathbf{I}$ is the matrix with $1$s on the diagonal and $0$s elsewhere.
So, I can write $\mathbf{M}$ as $-4 \mathbf{I}.$
To find $\mathbf{M},$ I just need to multiply each entry of $\mathbf{I}$ by $-4.$