d the following matrices where \( \mathrm{A}=\left[\begin{array}{rr}6 & -7 \\ -4 & 6 \\ -1 & 4\end{array}\right] \) and \( \mathrm{B}=\left[\begin{array}{rr}-9 & 8 \\ 3 & 0 \\ -3 & 8\end{array}\right] \) \( \begin{array}{lll}\text { a. } \mathrm{A}+\mathrm{B} & \text { b. } \mathrm{A}-\mathrm{B} & \text { d. }-3 A-9 B\end{array} \) A. \( \mathrm{A}+\mathrm{B}=\square \) (Simplify your answer.) B. This matrix operation is not possible:

To perform the matrix operations, we first need to check the dimensions of the matrices \( \mathrm{A} \) and \( \mathrm{B} \). Matrix \( \mathrm{A} \) is a \( 3 \times 2 \) matrix: \[ \mathrm{A} = \begin{bmatrix} 6 & -7 \\ -4 & 6 \\ -1 & 4 \end{bmatrix} \] Matrix \( \mathrm{B} \) is also a \( 3 \times 2 \) matrix: \[ \mathrm{B} = \begin{bmatrix} -9 & 8 \\ 3 & 0 \\ -3 & 8 \end{bmatrix} \] Since both matrices have the same dimensions, we can perform addition and subtraction. ### a. \( \mathrm{A} + \mathrm{B} \) To add the matrices, we add the corresponding elements: \[ \mathrm{A} + \mathrm{B} = \begin{bmatrix} 6 + (-9) & -7 + 8 \\ -4 + 3 & 6 + 0 \\ -1 + (-3) & 4 + 8 \end{bmatrix} \] Calculating each element: \[ = \begin{bmatrix} -3 & 1 \\ -1 & 6 \\ -4 & 12 \end{bmatrix} \] So, \[ \mathrm{A} + \mathrm{B} = \begin{bmatrix} -3 & 1 \\ -1 & 6 \\ -4 & 12 \end{bmatrix} \] ### b. \( \mathrm{A} - \mathrm{B} \) To subtract the matrices, we subtract the corresponding elements: \[ \mathrm{A} - \mathrm{B} = \begin{bmatrix} 6 - (-9) & -7 - 8 \\ -4 - 3 & 6 - 0 \\ -1 - (-3) & 4 - 8 \end{bmatrix} \] Calculating each element: \[ = \begin{bmatrix} 15 & -15 \\ -7 & 6 \\ 2 & -4 \end{bmatrix} \] So, \[ \mathrm{A} - \mathrm{B} = \begin{bmatrix} 15 & -15 \\ -7 & 6 \\ 2 & -4 \end{bmatrix} \] ### d. \( -3\mathrm{A} - 9\mathrm{B} \) First, we calculate \( -3\mathrm{A} \) and \( -9\mathrm{B} \): \[ -3\mathrm{A} = -3 \begin{bmatrix} 6 & -7 \\ -4 & 6 \\ -1 & 4 \end{bmatrix} = \begin{bmatrix} -18 & 21 \\ 12 & -18 \\ 3 & -12 \end{bmatrix} \] \[ -9\mathrm{B} = -9 \begin{bmatrix} -9 & 8 \\ 3 & 0 \\ -3 & 8 \end{bmatrix} = \begin{bmatrix} 81 & -72 \\ -27 & 0 \\ 27 & -72 \end{bmatrix} \] Now, we add these two results: \[ -3\mathrm{A} - 9\mathrm{B} = \begin{bmatrix} -18 & 21 \\ 12 & -18 \\ 3 & -12 \end{bmatrix} + \begin{bmatrix} 81 & -72 \\ -27 & 0 \\ 27 & -72 \end{bmatrix} \] Calculating each element: \[ = \begin{bmatrix} -18 + 81 & 21 - 72 \\ 12 - 27 & -18 + 0 \\ 3 + 27 & -12 - 72 \end{bmatrix} \] \[ = \begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix} \] So, \[ -3\mathrm{A} - 9\mathrm{B} = \begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix} \] ### Summary of Results: - \( \mathrm{A} + \mathrm{B} = \begin{bmatrix} -3 & 1 \\ -1 & 6 \\ -4 & 12 \end{bmatrix} \) - \( \mathrm{A} - \mathrm{B} = \begin{bmatrix} 15 & -15 \\ -7 & 6 \\ 2 & -4 \end{bmatrix} \) - \( -3\mathrm{A} - 9\mathrm{B} = \begin{bmatrix} 63 & -51 \\ -15 & -18 \\ 30 & -84 \end{bmatrix} \)