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(D) Given $A = \begin{bmatrix} 7 & 3 & \alpha \ \beta & 1 & -11 \ -5 & \gamma & 19 \end{bmatrix}$ and $A \begin{bmatrix} 5 \ -13 \ 11 \end{bmatrix

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$-2A$

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(D) Given $A = \begin{bmatrix} 7 & 3 & \alpha \ \beta & 1 & -11 \ -5 & \gamma & 19 \end{bmatrix}$ and $A \begin{bmatrix} 5 \ -13 \ 11 \end{bmatrix} = \begin{bmatrix} -290 \ -119 \ 210 \end{bmatrix}$.Performing matrix multiplication:$35 - 39 + 11\alpha = -290 \Rightarrow 11\alpha = -286 \Rightarrow \alpha = -26$.$5\beta - 13 - 121 = -119 \Rightarrow 5\beta = 15 \Rightarrow \beta = 3$.$-25 - 13\gamma + 209 = 210 \Rightarrow -13\gamma = 26 \Rightarrow \gamma = -2$.Thus,$A = \begin{bmatrix} 7 & 3 & -26 \ 3 & 1 & -11 \ -5 & -2 & 19 \end{bmatrix}$.Calculating the determinant $|A| = 7(19 - 22) - 3(57 - 55) - 26(-6 + 5) = 7(-3) - 3(2) - 26(-1) = -21 - 6 + 26 = -1$.We know that $(\operatorname{adj} A)^{-1} = \frac{A}{|A|} = -A$ and $\operatorname{adj} A^{-1} = \operatorname{adj}(\frac{\operatorname{adj} A}{|A|}) = \frac{1}{|A|^{n-1}} \operatorname{adj}(\operatorname{adj} A) = \frac{1}{(-1)^2} |A| A = -A$.Therefore,$(\operatorname{adj} A)^{-1} + \operatorname{adj} A^{-1} = -A + (-A) = -2A$.

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