Let $\mathbf{A}$ and $\mathbf{B}$ be $2 \times 2$ matrices such that $\det \mathbf{A} = -1$ and $\det \mathbf{B} = 3.$  Find $\det (3 \mathbf{A} \mathbf{B}).$
One property is that $\det (\mathbf{A} \mathbf{B}) = \det \mathbf{A} \det \mathbf{B}$ for any square matrices $\mathbf{A}$ and $\mathbf{B}.$ Another property is that $\det (c \mathbf{A}) = c^n \det \mathbf{A}$ for any scalar $c$ and $n \times n$ matrix $\mathbf{A}.$ Using these properties, I can write $\det (3 \mathbf{A} \mathbf{B}) = \det (3 \mathbf{A}) \det \mathbf{B} = 3^2 \det \mathbf{A} \det \mathbf{B} = 9 \cdot (-1) \cdot 3 = -27.$ 