1. Problem: Find all minors and cofactors of matrix \(A = \begin{bmatrix}1 & -2 & 3 \\ 6 & 7 & -1 \\ -3 & 1 & 4\end{bmatrix}\). 2. The minor \(M_{ij}\) of an element is the determinant of the matrix formed by deleting the \(i\)-th row and \(j\)-th column. 3. The cofactor \(C_{ij} = (-1)^{i+j} M_{ij}\). 4. Calculate minors and cofactors for each element: - \(M_{11} = \det \begin{bmatrix}7 & -1 \\ 1 & 4\end{bmatrix} = 7 \times 4 - (-1) \times 1 = 28 + 1 = 29\) - \(C_{11} = (+1) \times 29 = 29\) - \(M_{12} = \det \begin{bmatrix}6 & -1 \\ -3 & 4\end{bmatrix} = 6 \times 4 - (-1) \times (-3) = 24 - 3 = 21\) - \(C_{12} = (-1) \times 21 = -21\) - \(M_{13} = \det \begin{bmatrix}6 & 7 \\ -3 & 1\end{bmatrix} = 6 \times 1 - 7 \times (-3) = 6 + 21 = 27\) - \(C_{13} = (+1) \times 27 = 27\) - \(M_{21} = \det \begin{bmatrix}-2 & 3 \\ 1 & 4\end{bmatrix} = (-2) \times 4 - 3 \times 1 = -8 - 3 = -11\) - \(C_{21} = (-1) \times (-11) = 11\) - \(M_{22} = \det \begin{bmatrix}1 & 3 \\ -3 & 4\end{bmatrix} = 1 \times 4 - 3 \times (-3) = 4 + 9 = 13\) - \(C_{22} = (+1) \times 13 = 13\) - \(M_{23} = \det \begin{bmatrix}1 & -2 \\ -3 & 1\end{bmatrix} = 1 \times 1 - (-2) \times (-3) = 1 - 6 = -5\) - \(C_{23} = (-1) \times (-5) = 5\) - \(M_{31} = \det \begin{bmatrix}-2 & 3 \\ 7 & -1\end{bmatrix} = (-2) \times (-1) - 3 \times 7 = 2 - 21 = -19\) - \(C_{31} = (+1) \times (-19) = -19\) - \(M_{32} = \det \begin{bmatrix}1 & 3 \\ 6 & -1\end{bmatrix} = 1 \times (-1) - 3 \times 6 = -1 - 18 = -19\) - \(C_{32} = (-1) \times (-19) = 19\) - \(M_{33} = \det \begin{bmatrix}1 & -2 \\ 6 & 7\end{bmatrix} = 1 \times 7 - (-2) \times 6 = 7 + 12 = 19\) - \(C_{33} = (+1) \times 19 = 19\) Final answer: Minors and cofactors as above for matrix 1. q_count: 1