Matrix Inverse 9Bd2Df

1. **State the problem:** Find the inverse of the matrix $$\mathbf{A}=\begin{bmatrix}2 & -17 & 11 \\ -11 & -7 & 0 \\ 3 & -2 & 1\end{bmatrix}.$$\n\n2. **Recall the formula for the inverse of a 3x3 matrix:**\n$$\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \mathrm{adj}(\mathbf{A}),$$\nwhere $\det(\mathbf{A})$ is the determinant of $\mathbf{A}$ and $\mathrm{adj}(\mathbf{A})$ is the adjugate matrix (transpose of the cofactor matrix).\n\n3. **Calculate the determinant $\det(\mathbf{A})$:}\n$$\det(\mathbf{A}) = 2 \begin{vmatrix} -7 & 0 \\ -2 & 1 \end{vmatrix} - (-17) \begin{vmatrix} -11 & 0 \\ 3 & 1 \end{vmatrix} + 11 \begin{vmatrix} -11 & -7 \\ 3 & -2 \end{vmatrix}.$$\nCalculate each minor:\n$$\begin{vmatrix} -7 & 0 \\ -2 & 1 \end{vmatrix} = (-7)(1) - (0)(-2) = -7,$$\n$$\begin{vmatrix} -11 & 0 \\ 3 & 1 \end{vmatrix} = (-11)(1) - (0)(3) = -11,$$\n$$\begin{vmatrix} -11 & -7 \\ 3 & -2 \end{vmatrix} = (-11)(-2) - (-7)(3) = 22 + 21 = 43.$$\nSubstitute back:\n$$\det(\mathbf{A}) = 2(-7) - (-17)(-11) + 11(43) = -14 - 187 + 473 = 272.$$\n\n4. **Calculate the cofactor matrix:**\nEach cofactor $C_{ij} = (-1)^{i+j} M_{ij}$ where $M_{ij}$ is the minor of element in row $i$, column $j$.\n\n- $C_{11} = \begin{vmatrix} -7 & 0 \\ -2 & 1 \end{vmatrix} = -7$\n- $C_{12} = - \begin{vmatrix} -11 & 0 \\ 3 & 1 \end{vmatrix} = -(-11) = 11$\n- $C_{13} = \begin{vmatrix} -11 & -7 \\ 3 & -2 \end{vmatrix} = 43$\n\n- $C_{21} = - \begin{vmatrix} -17 & 11 \\ -2 & 1 \end{vmatrix} = -((-17)(1) - (11)(-2)) = -(-17 + 22) = -5$\n- $C_{22} = \begin{vmatrix} 2 & 11 \\ 3 & 1 \end{vmatrix} = 2(1) - 11(3) = 2 - 33 = -31$\n- $C_{23} = - \begin{vmatrix} 2 & -17 \\ 3 & -2 \end{vmatrix} = - (2(-2) - (-17)(3)) = -(-4 + 51) = -47$\n\n- $C_{31} = \begin{vmatrix} -17 & 11 \\ -7 & 0 \end{vmatrix} = (-17)(0) - (11)(-7) = 77$\n- $C_{32} = - \begin{vmatrix} 2 & 11 \\ -11 & 0 \end{vmatrix} = - (2(0) - 11(-11)) = - (0 + 121) = -121$\n- $C_{33} = \begin{vmatrix} 2 & -17 \\ -11 & -7 \end{vmatrix} = 2(-7) - (-17)(-11) = -14 - 187 = -201$\n\n5. **Form the cofactor matrix:**\n$$\mathrm{Cof}(\mathbf{A}) = \begin{bmatrix} -7 & 11 & 43 \\ -5 & -31 & -47 \\ 77 & -121 & -201 \end{bmatrix}.$$\n\n6. **Find the adjugate matrix by transposing the cofactor matrix:**\n$$\mathrm{adj}(\mathbf{A}) = \begin{bmatrix} -7 & -5 & 77 \\ 11 & -31 & -121 \\ 43 & -47 & -201 \end{bmatrix}.$$\n\n7. **Calculate the inverse matrix:**\n$$\mathbf{A}^{-1} = \frac{1}{272} \begin{bmatrix} -7 & -5 & 77 \\ 11 & -31 & -121 \\ 43 & -47 & -201 \end{bmatrix}.$$\n\n**Final answer:**\n$$\boxed{\mathbf{A}^{-1} = \frac{1}{272} \begin{bmatrix} -7 & -5 & 77 \\ 11 & -31 & -121 \\ 43 & -47 & -201 \end{bmatrix}}.$$