Matrix Inverse 67F546

1. **State the problem:** Find the inverse of the matrix $$A=\begin{bmatrix}2 & -17 & 1 \\ 1 & -1 & 1 \\ -7 & 0 & 3 \end{bmatrix}$$. 2. **Recall the formula:** The inverse of a matrix $$A$$ exists if $$\det(A) \neq 0$$ and is given by $$A^{-1} = \frac{1}{\det(A)} \mathrm{adj}(A)$$ where $$\mathrm{adj}(A)$$ is the adjugate matrix. 3. **Calculate the determinant $$\det(A)$$:** $$\det(A) = 2 \begin{vmatrix} -1 & 1 \\ 0 & 3 \end{vmatrix} - (-17) \begin{vmatrix} 1 & 1 \\ -7 & 3 \end{vmatrix} + 1 \begin{vmatrix} 1 & -1 \\ -7 & 0 \end{vmatrix}$$ Calculate each minor: $$\begin{vmatrix} -1 & 1 \\ 0 & 3 \end{vmatrix} = (-1)(3) - (1)(0) = -3$$ $$\begin{vmatrix} 1 & 1 \\ -7 & 3 \end{vmatrix} = (1)(3) - (1)(-7) = 3 + 7 = 10$$ $$\begin{vmatrix} 1 & -1 \\ -7 & 0 \end{vmatrix} = (1)(0) - (-1)(-7) = 0 - 7 = -7$$ Substitute back: $$\det(A) = 2(-3) - (-17)(10) + 1(-7) = -6 + 170 - 7 = 157$$ Since $$\det(A) = 157 \neq 0$$, the inverse exists. 4. **Find the matrix of cofactors:** Calculate each cofactor $$C_{ij} = (-1)^{i+j} M_{ij}$$ where $$M_{ij}$$ is the minor of element $$a_{ij}$$. - $$C_{11} = (+1) \times \begin{vmatrix} -1 & 1 \\ 0 & 3 \end{vmatrix} = -3$$ - $$C_{12} = (-1) \times \begin{vmatrix} 1 & 1 \\ -7 & 3 \end{vmatrix} = -10$$ - $$C_{13} = (+1) \times \begin{vmatrix} 1 & -1 \\ -7 & 0 \end{vmatrix} = -7$$ - $$C_{21} = (-1) \times \begin{vmatrix} -17 & 1 \\ 0 & 3 \end{vmatrix} = -(-17 \times 3 - 1 \times 0) = -(-51) = 51$$ - $$C_{22} = (+1) \times \begin{vmatrix} 2 & 1 \\ -7 & 3 \end{vmatrix} = 2 \times 3 - 1 \times (-7) = 6 + 7 = 13$$ - $$C_{23} = (-1) \times \begin{vmatrix} 2 & -17 \\ -7 & 0 \end{vmatrix} = - (2 \times 0 - (-17) \times (-7)) = - (0 - 119) = 119$$ - $$C_{31} = (+1) \times \begin{vmatrix} -17 & 1 \\ -1 & 1 \end{vmatrix} = (-17)(1) - (1)(-1) = -17 + 1 = -16$$ - $$C_{32} = (-1) \times \begin{vmatrix} 2 & 1 \\ 1 & 1 \end{vmatrix} = - (2 \times 1 - 1 \times 1) = - (2 - 1) = -1$$ - $$C_{33} = (+1) \times \begin{vmatrix} 2 & -17 \\ 1 & -1 \end{vmatrix} = 2 \times (-1) - (-17) \times 1 = -2 + 17 = 15$$ 5. **Form the cofactor matrix:** $$\mathrm{Cof}(A) = \begin{bmatrix} -3 & -10 & -7 \\ 51 & 13 & 119 \\ -16 & -1 & 15 \end{bmatrix}$$ 6. **Find the adjugate matrix $$\mathrm{adj}(A)$$ by transposing the cofactor matrix:** $$\mathrm{adj}(A) = \begin{bmatrix} -3 & 51 & -16 \\ -10 & 13 & -1 \\ -7 & 119 & 15 \end{bmatrix}$$ 7. **Calculate the inverse:** $$A^{-1} = \frac{1}{157} \begin{bmatrix} -3 & 51 & -16 \\ -10 & 13 & -1 \\ -7 & 119 & 15 \end{bmatrix}$$ **Final answer:** $$\boxed{A^{-1} = \frac{1}{157} \begin{bmatrix} -3 & 51 & -16 \\ -10 & 13 & -1 \\ -7 & 119 & 15 \end{bmatrix}}$$