### Instruction: Find the matrix $\mathbf{M}$ such that
\[\mathbf{M} \mathbf{v} = -5 \mathbf{v}\]for all vectors $\mathbf{v}.$.### Response: I want to find a matrix that multiplies any vector by -5.
That means the matrix should have -5 on the main diagonal and 0 everywhere else.
In other words, it should be a scalar multiple of the identity matrix.
The identity matrix has 1 on the main diagonal and 0 everywhere else.
So, to get -5 on the main diagonal, I just need to multiply the identity matrix by -5.
The matrix I'm looking for is -5 times the identity matrix.
I can write that as -5$\mathbf{I}$, where $\mathbf{I}$ is the identity matrix.