1. **Problem:** Given matrices $P = \begin{pmatrix}3 & -4 \\ 5 & 1\end{pmatrix}$ and $Q = \begin{pmatrix}2 \\ 8\end{pmatrix}$, find $P^{-1}$. 2. **Formula for inverse of a 2x2 matrix:** For $P = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$, $$P^{-1} = \frac{1}{\det(P)} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$$ where $\det(P) = ad - bc$. 3. **Calculate determinant of $P$:** $$\det(P) = (3)(1) - (5)(-4) = 3 + 20 = 23$$ 4. **Calculate inverse matrix:** $$P^{-1} = \frac{1}{23} \begin{pmatrix}1 & 4 \\ -5 & 3\end{pmatrix} = \begin{pmatrix}\frac{1}{23} & \frac{4}{23} \\ -\frac{5}{23} & \frac{3}{23}\end{pmatrix}$$ 5. **Explanation:** The inverse exists because $\det(P) \neq 0$. We swap the diagonal elements, change signs of off-diagonal elements, then divide by determinant. **Final answer:** $$P^{-1} = \begin{pmatrix}\frac{1}{23} & \frac{4}{23} \\ -\frac{5}{23} & \frac{3}{23}\end{pmatrix}$$