1. **State the problem:** Calculate the determinant of the matrix $$A = \begin{pmatrix} 2 & 1 & -1 \\ 0 & 4 & 3 \\ -1 & 6 & 0 \end{pmatrix}$$ and then find its inverse $$A^{-1}$$. 2. **Formula for determinant of a 3x3 matrix:** $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$. 3. **Calculate the determinant:** Here, $$a=2, b=1, c=-1, d=0, e=4, f=3, g=-1, h=6, i=0$$. Calculate each minor: $$ei - fh = 4 \times 0 - 3 \times 6 = 0 - 18 = -18$$ $$di - fg = 0 \times 0 - 3 \times (-1) = 0 + 3 = 3$$ $$dh - eg = 0 \times 6 - 4 \times (-1) = 0 + 4 = 4$$ So, $$\det(A) = 2(-18) - 1(3) + (-1)(4) = -36 - 3 - 4 = -43$$ 4. **Since $$\det(A) \neq 0$$, the inverse exists.** 5. **Formula for inverse:** $$A^{-1} = \frac{1}{\det(A)} \mathrm{adj}(A)$$ where $$\mathrm{adj}(A)$$ is the adjugate matrix, the transpose of the cofactor matrix. 6. **Calculate cofactors:** - $$C_{11} = +(ei - fh) = -18$$ - $$C_{12} = -(di - fg) = -3$$ - $$C_{13} = +(dh - eg) = 4$$ - $$C_{21} = -(bi - ch) = -(1 \times 0 - (-1) \times 6) = -(0 + 6) = -6$$ - $$C_{22} = +(ai - cg) = 2 \times 0 - (-1) \times (-1) = 0 - 1 = -1$$ - $$C_{23} = -(ah - bg) = -(2 \times 6 - 1 \times (-1)) = -(12 + 1) = -13$$ - $$C_{31} = +(bf - ce) = 1 \times 3 - (-1) \times 4 = 3 + 4 = 7$$ - $$C_{32} = -(af - cd) = -(2 \times 3 - (-1) \times 0) = -(6 - 0) = -6$$ - $$C_{33} = +(ae - bd) = 2 \times 4 - 1 \times 0 = 8 - 0 = 8$$ 7. **Cofactor matrix:** $$\begin{pmatrix} -18 & -3 & 4 \\ -6 & -1 & -13 \\ 7 & -6 & 8 \end{pmatrix}$$ 8. **Adjugate matrix (transpose of cofactor matrix):** $$\mathrm{adj}(A) = \begin{pmatrix} -18 & -6 & 7 \\ -3 & -1 & -6 \\ 4 & -13 & 8 \end{pmatrix}$$ 9. **Calculate inverse:** $$A^{-1} = \frac{1}{-43} \begin{pmatrix} -18 & -6 & 7 \\ -3 & -1 & -6 \\ 4 & -13 & 8 \end{pmatrix} = \begin{pmatrix} \frac{18}{43} & \frac{6}{43} & -\frac{7}{43} \\ \frac{3}{43} & \frac{1}{43} & \frac{6}{43} \\ -\frac{4}{43} & \frac{13}{43} & -\frac{8}{43} \end{pmatrix}$$ **Final answers:** - $$\det(A) = -43$$ - $$A^{-1} = \begin{pmatrix} \frac{18}{43} & \frac{6}{43} & -\frac{7}{43} \\ \frac{3}{43} & \frac{1}{43} & \frac{6}{43} \\ -\frac{4}{43} & \frac{13}{43} & -\frac{8}{43} \end{pmatrix}$$