If $\mathbf{A}$ is a $2 \times 2$ matrix such that $\det \mathbf{A} = 2,$ then find $\det (\mathbf{-3A}).$
I know that for any square matrix $\mathbf{A},$ the determinant of a scalar multiple $k \mathbf{A}$ is $k^n \det \mathbf{A},$ where $n$ is the size of the matrix. So in this case, $\det (\mathbf{-3A}) = (-3)^2 \det \mathbf{A}.$ I also know that $\det \mathbf{A} = 2,$ so I can substitute that in. Therefore, $\det (\mathbf{-3A}) = (-3)^2 \cdot 2 = 9 \cdot 2 = 18.$ 