### Instruction: If the matrix $\mathbf{A}$ has an inverse and $(\mathbf{A} - 2 \mathbf{I})(\mathbf{A} - 4 \mathbf{I}) = \mathbf{0},$ then find
\[\mathbf{A} + 8 \mathbf{A}^{-1}.\].### Response: Let's expand the given equation.
That gives $(\mathbf{A} - 2 \mathbf{I})(\mathbf{A} - 4 \mathbf{I}) = \mathbf{A}^2 - 6 \mathbf{A} + 8 \mathbf{I} = \mathbf{0}.$
Right, and since $\mathbf{A}$ has an inverse, we can multiply both sides of this equation by $\mathbf{A}^{-1}$ to get $\mathbf{A} - 6 \mathbf{I} + 8 \mathbf{A}^{-1} = \mathbf{0}.$