\( \left. \begin{array} { c c } { - A = \left( \begin{array} { c c c } { 3 } & { 1 } & { 3 } \\ { - 1 } & { 0 } & { - 1 } \\ { 2 } & { 5 } & { 3 } \end{array} \right) } & { B = \left( \begin{array} { c c c } { 7 } & { 3 } & { - 1 } \\ { 2 } & { 8 } & { 7 } \\ { - 1 } & { - 2 } & { - 3 } \end{array} \right) } \\ { A \times B } & { A + B } \\ { B \times A } & { B - A } \end{array} \right. \)

To solve the problem, we need to perform matrix operations involving the matrices \( A \) and \( B \). First, we need to find the matrix \( A \) from the given information that \( -A \) is provided. Given: \[ -A = \begin{pmatrix} 3 & 1 & 3 \\ -1 & 0 & -1 \\ 2 & 5 & 3 \end{pmatrix} \] Thus, we can find \( A \) by multiplying by -1: \[ A = -\begin{pmatrix} 3 & 1 & 3 \\ -1 & 0 & -1 \\ 2 & 5 & 3 \end{pmatrix} = \begin{pmatrix} -3 & -1 & -3 \\ 1 & 0 & 1 \\ -2 & -5 & -3 \end{pmatrix} \] Now we have: \[ A = \begin{pmatrix} -3 & -1 & -3 \\ 1 & 0 & 1 \\ -2 & -5 & -3 \end{pmatrix} \] \[ B = \begin{pmatrix} 7 & 3 & -1 \\ 2 & 8 & 7 \\ -1 & -2 & -3 \end{pmatrix} \] ### 1. Calculate \( A \times B \) To multiply \( A \) and \( B \): \[ A \times B = \begin{pmatrix} -3 & -1 & -3 \\ 1 & 0 & 1 \\ -2 & -5 & -3 \end{pmatrix} \times \begin{pmatrix} 7 & 3 & -1 \\ 2 & 8 & 7 \\ -1 & -2 & -3 \end{pmatrix} \] Calculating each element of the resulting matrix: - First row: - \( (-3)(7) + (-1)(2) + (-3)(-1) = -21 - 2 + 3 = -20 \) - \( (-3)(3) + (-1)(8) + (-3)(-2) = -9 - 8 + 6 = -11 \) - \( (-3)(-1) + (-1)(7) + (-3)(-3) = 3 - 7 + 9 = 5 \) - Second row: - \( (1)(7) + (0)(2) + (1)(-1) = 7 + 0 - 1 = 6 \) - \( (1)(3) + (0)(8) + (1)(-2) = 3 + 0 - 2 = 1 \) - \( (1)(-1) + (0)(7) + (1)(-3) = -1 + 0 - 3 = -4 \) - Third row: - \( (-2)(7) + (-5)(2) + (-3)(-1) = -14 - 10 + 3 = -21 \) - \( (-2)(3) + (-5)(8) + (-3)(-2) = -6 - 40 + 6 = -40 \) - \( (-2)(-1) + (-5)(7) + (-3)(-3) = 2 - 35 + 9 = -24 \) Thus, we have: \[ A \times B = \begin{pmatrix} -20 & -11 & 5 \\ 6 & 1 & -4 \\ -21 & -40 & -24 \end{pmatrix} \] ### 2. Calculate \( A + B \) Now, we add \( A \) and \( B \): \[ A + B = \begin{pmatrix} -3 & -1 & -3 \\ 1 & 0 & 1 \\ -2 & -5 & -3 \end{pmatrix} + \begin{pmatrix} 7 & 3 & -1 \\ 2 & 8 & 7 \\ -1 & -2 & -3 \end{pmatrix} \] Calculating each element: - First row: \( -3 + 7 = 4 \), \( -1 + 3 = 2 \), \( -3 - 1 = -4 \) - Second row: \( 1 + 2 = 3 \), \( 0 + 8 = 8 \), \( 1 + 7 = 8 \) - Third row: \( -2 - 1 = -3 \), \( -5 - 2 = -7 \