1. **State the problem:** Find the determinant and inverse of matrix $$A=\begin{bmatrix}5 & -7 & 1 \\ 6 & -8 & -2 \\ 3 & -1 & 6\end{bmatrix}$$. 2. **Find the determinant of matrix A:** The determinant of a 3x3 matrix $$A=\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$ is given by: $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. **Apply the formula:** $$\det(A) = 5((-8)(6) - (-2)(-1)) - (-7)(6 \cdot 6 - (-2) \cdot 3) + 1(6 \cdot (-1) - (-8) \cdot 3)$$ 4. **Calculate each term:** $$5((-48) - 2) + 7(36 + 6) + 1(-6 + 24)$$ 5. **Simplify:** $$5(-50) + 7(42) + 1(18) = -250 + 294 + 18$$ 6. **Sum the results:** $$-250 + 294 + 18 = 62$$ 7. **Determinant result:** $$\det(A) = 62$$ 8. **Find the inverse of matrix A:** The inverse exists because $$\det(A) \neq 0$$. 9. **Formula for inverse:** $$A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)$$ where adj(A) is the adjugate matrix. 10. **Calculate the matrix of minors:** - Minor of element (1,1): $$\det\begin{bmatrix}-8 & -2 \\ -1 & 6\end{bmatrix} = (-8)(6) - (-2)(-1) = -48 - 2 = -50$$ - Minor of element (1,2): $$\det\begin{bmatrix}6 & -2 \\ 3 & 6\end{bmatrix} = 6 \cdot 6 - (-2) \cdot 3 = 36 + 6 = 42$$ - Minor of element (1,3): $$\det\begin{bmatrix}6 & -8 \\ 3 & -1\end{bmatrix} = 6(-1) - (-8)(3) = -6 + 24 = 18$$ - Minor of element (2,1): $$\det\begin{bmatrix}-7 & 1 \\ -1 & 6\end{bmatrix} = (-7)(6) - 1(-1) = -42 + 1 = -41$$ - Minor of element (2,2): $$\det\begin{bmatrix}5 & 1 \\ 3 & 6\end{bmatrix} = 5 \cdot 6 - 1 \cdot 3 = 30 - 3 = 27$$ - Minor of element (2,3): $$\det\begin{bmatrix}5 & -7 \\ 3 & -1\end{bmatrix} = 5(-1) - (-7)(3) = -5 + 21 = 16$$ - Minor of element (3,1): $$\det\begin{bmatrix}-7 & 1 \\ -8 & -2\end{bmatrix} = (-7)(-2) - 1(-8) = 14 + 8 = 22$$ - Minor of element (3,2): $$\det\begin{bmatrix}5 & 1 \\ 6 & -2\end{bmatrix} = 5(-2) - 1(6) = -10 - 6 = -16$$ - Minor of element (3,3): $$\det\begin{bmatrix}5 & -7 \\ 6 & -8\end{bmatrix} = 5(-8) - (-7)(6) = -40 + 42 = 2$$ 11. **Form the matrix of minors:** $$\begin{bmatrix}-50 & 42 & 18 \\ -41 & 27 & 16 \\ 22 & -16 & 2\end{bmatrix}$$ 12. **Apply cofactor signs:** Multiply each element by $$(-1)^{i+j}$$: $$\begin{bmatrix}-50 & -42 & 18 \\ 41 & 27 & -16 \\ 22 & 16 & 2\end{bmatrix}$$ 13. **Transpose to get adjugate matrix:** $$\text{adj}(A) = \begin{bmatrix}-50 & 41 & 22 \\ -42 & 27 & 16 \\ 18 & -16 & 2\end{bmatrix}$$ 14. **Calculate inverse:** $$A^{-1} = \frac{1}{62} \cdot \begin{bmatrix}-50 & 41 & 22 \\ -42 & 27 & 16 \\ 18 & -16 & 2\end{bmatrix}$$ 15. **Final answer:** $$\boxed{\det(A) = 62}$$ $$\boxed{A^{-1} = \frac{1}{62} \begin{bmatrix}-50 & 41 & 22 \\ -42 & 27 & 16 \\ 18 & -16 & 2\end{bmatrix}}$$