1. **Problem Statement:** Find the inverse of matrix $$P = \begin{pmatrix}8 & 5 \\ 6 & 9\end{pmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$$, the inverse $$A^{-1}$$ is given by: $$ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$ provided that the determinant $$ad - bc \neq 0$$. 3. **Calculate the determinant of P:** $$ \det(P) = (8)(9) - (5)(6) = 72 - 30 = 42 $$ Since $$42 \neq 0$$, the inverse exists. 4. **Apply the formula:** $$ P^{-1} = \frac{1}{42} \begin{pmatrix} 9 & -5 \\ -6 & 8 \end{pmatrix} = \begin{pmatrix} \frac{9}{42} & \frac{-5}{42} \\ \frac{-6}{42} & \frac{8}{42} \end{pmatrix} $$ 5. **Simplify fractions:** $$ P^{-1} = \begin{pmatrix} \frac{3}{14} & -\frac{5}{42} \\ -\frac{1}{7} & \frac{4}{21} \end{pmatrix} $$ **Final answer:** $$ P^{-1} = \begin{pmatrix} \frac{3}{14} & -\frac{5}{42} \\ -\frac{1}{7} & \frac{4}{21} \end{pmatrix} $$