Find the inverse of \( \quad A=\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8\end{array}\right] \)

To find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{bmatrix} \), we can use the formula for the inverse of a matrix, which is given by \( A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A) \), where \( \text{det}(A) \) is the determinant of \( A \) and \( \text{adj}(A) \) is the adjugate of \( A \). First, we calculate the determinant of \( A \): \[ \text{det}(A) = 1 \cdot (5 \cdot 8 - 3 \cdot 0) - 2 \cdot (2 \cdot 8 - 3 \cdot 1) + 3 \cdot (2 \cdot 0 - 5 \cdot 1) \] \[ = 1 \cdot (40 - 0) - 2 \cdot (16 - 3) + 3 \cdot (0 - 5) \] \[ = 40 - 2 \cdot 13 - 15 \] \[ = 40 - 26 - 15 = -1. \] Now, we know that \( \text{det}(A) = -1 \). Next, we need to find the adjugate of \( A \), which involves finding the cofactor matrix and then transposing it. The cofactor of each element is computed by taking the determinant of the 2x2 matrix that remains when the row and column of that element are removed. Calculating the cofactors: 1. For \( a_{11} = 1 \), the cofactor is \( \text{det}\begin{bmatrix} 5 & 3 \\ 0 & 8 \end{bmatrix} = 5 \cdot 8 - 3 \cdot 0 = 40 \). 2. For \( a_{12} = 2 \), the cofactor is \( -\text{det}\begin{bmatrix} 2 & 3 \\ 1 & 8 \end{bmatrix} = -(2 \cdot 8 - 3 \cdot 1) = -(16 - 3) = -13 \). 3. For \( a_{13} = 3 \), the cofactor is \( \text{det}\begin{bmatrix} 2 & 5 \\ 1 & 0 \end{bmatrix} = 2 \cdot 0 - 5 \cdot 1 = -5 \). 4. For \( a_{21} = 2 \), the cofactor is \( -\text{det}\begin{bmatrix} 2 & 3 \\ 0 & 8 \end{bmatrix} = -(2 \cdot 8 - 3 \cdot 0) = -16 \). 5. For \( a_{22} = 5 \), the cofactor is \( \text{det}\begin{bmatrix} 1 & 3 \\ 1 & 8 \end{bmatrix} = 1 \cdot 8 - 3 \cdot 1 = 5 \). 6. For \( a_{23} = 3 \), the cofactor is \( -\text{det}\begin{bmatrix} 1 & 2 \\ 1 & 0 \end{bmatrix} = -(1 \cdot 0 - 2 \cdot 1) = 2 \). 7. For \( a_{31} = 1 \), the cofactor is \( \text{det}\begin{bmatrix} 2 & 3 \\ 5 & 3 \end{bmatrix} = 2 \cdot 3 - 3 \cdot 5 = 6 - 15 = -9 \). 8. For \( a_{32} = 0 \), the cofactor is \( -\text{det}\begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix} = -(1 \cdot 3 - 3 \cdot 2) = -(-3) = 3 \). 9. For \( a_{33} = 8 \), the cofactor is \( \text{det}\begin{bmatrix} 1 & 2 \\ 2 & 5 \end{bmatrix} = 1 \cdot 5 - 2 \cdot 2 = 5 -