What is the inverse of A? \( \left[\begin{array}{ccc}1 & 3 & 1 \\ 2 & 7 & 4 \\ 1 & -1 & -5\end{array}\right] \)

To find the inverse of the matrix \[ A = \begin{bmatrix} 1 & 3 & 1 \\ 2 & 7 & 4 \\ 1 & -1 & -5 \end{bmatrix}, \] we can follow these steps: ### 1. Calculate the Determinant of \( A \) The determinant of a \( 3 \times 3 \) matrix \[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \] is given by: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg). \] For matrix \( A \): \[ \text{det}(A) = 1(7 \times (-5) - 4 \times (-1)) - 3(2 \times (-5) - 4 \times 1) + 1(2 \times (-1) - 7 \times 1) \] \[ = 1(-35 + 4) - 3(-10 - 4) + 1(-2 - 7) \] \[ = -31 + 42 - 9 = 2. \] Since \(\text{det}(A) \neq 0\), the inverse exists. ### 2. Find the Matrix of Minors Calculate the minor for each element by determining the determinant of the \( 2 \times 2 \) submatrix that remains after removing the row and column of that element. \[ \text{Minor}(A) = \begin{bmatrix} -31 & -14 & -9 \\ 14 & -6 & 4 \\ 5 & -2 & 1 \end{bmatrix}. \] ### 3. Apply Cofactors Apply the checkerboard pattern of signs to the matrix of minors to get the cofactor matrix: \[ \text{Cofactor}(A) = \begin{bmatrix} -31 & 14 & -9 \\ 14 & -6 & 4 \\ 5 & -2 & 1 \end{bmatrix}. \] ### 4. Transpose the Cofactor Matrix to Get the Adjugate \[ \text{Adj}(A) = \begin{bmatrix} -31 & 14 & 5 \\ 14 & -6 & -2 \\ -9 & 4 & 1 \end{bmatrix}. \] ### 5. Multiply by \( \frac{1}{\text{det}(A)} \) to Find the Inverse \[ A^{-1} = \frac{1}{2} \begin{bmatrix} -31 & 14 & 5 \\ 14 & -6 & -2 \\ -9 & 4 & 1 \end{bmatrix} = \begin{bmatrix} -\dfrac{31}{2} & 7 & \dfrac{5}{2} \\ 7 & -3 & -1 \\ -\dfrac{9}{2} & 2 & \dfrac{1}{2} \end{bmatrix}. \] ### Final Answer \[ A^{-1} = \begin{bmatrix} -\dfrac{31}{2} & 7 & \dfrac{5}{2} \\ 7 & -3 & -1 \\ -\dfrac{9}{2} & 2 & \dfrac{1}{2} \end{bmatrix} \]