1. **State the problem:** We are given two matrices $A$ and $B$ and asked to perform matrix operations: $-3A$, $A - 3B$, and $4A + 5B$. 2. **Recall matrix operations:** - Scalar multiplication: Multiply each element of the matrix by the scalar. - Matrix addition/subtraction: Add or subtract corresponding elements. 3. **Calculate $-3A$:** Given $A = \begin{bmatrix}-8 & 5 & 4 \\ 7 & 4 & 5 \\ 5 & -6 & 4\end{bmatrix}$, $$-3A = -3 \times A = \begin{bmatrix}-3 \times (-8) & -3 \times 5 & -3 \times 4 \\ -3 \times 7 & -3 \times 4 & -3 \times 5 \\ -3 \times 5 & -3 \times (-6) & -3 \times 4\end{bmatrix} = \begin{bmatrix}24 & -15 & -12 \\ -21 & -12 & -15 \\ -15 & 18 & -12\end{bmatrix}$$ 4. **Calculate $A - 3B$:** Given $B = \begin{bmatrix}0 & 1 & -4 \\ -5 & 5 & 1 \\ 4 & 5 & -8\end{bmatrix}$, First compute $3B$: $$3B = 3 \times B = \begin{bmatrix}0 & 3 & -12 \\ -15 & 15 & 3 \\ 12 & 15 & -24\end{bmatrix}$$ Then subtract: $$A - 3B = \begin{bmatrix}-8 - 0 & 5 - 3 & 4 - (-12) \\ 7 - (-15) & 4 - 15 & 5 - 3 \\ 5 - 12 & -6 - 15 & 4 - (-24)\end{bmatrix} = \begin{bmatrix}-8 & 2 & 16 \\ 22 & -11 & 2 \\ -7 & -21 & 28\end{bmatrix}$$ 5. **Calculate $4A + 5B$:** Compute $4A$: $$4A = 4 \times A = \begin{bmatrix}-32 & 20 & 16 \\ 28 & 16 & 20 \\ 20 & -24 & 16\end{bmatrix}$$ Compute $5B$: $$5B = 5 \times B = \begin{bmatrix}0 & 5 & -20 \\ -25 & 25 & 5 \\ 20 & 25 & -40\end{bmatrix}$$ Add $4A + 5B$: $$4A + 5B = \begin{bmatrix}-32 + 0 & 20 + 5 & 16 + (-20) \\ 28 + (-25) & 16 + 25 & 20 + 5 \\ 20 + 20 & -24 + 25 & 16 + (-40)\end{bmatrix} = \begin{bmatrix}-32 & 25 & -4 \\ 3 & 41 & 25 \\ 40 & 1 & -24\end{bmatrix}$$ Note: The user-provided matrix for $4A + 5B$ has the last row as $[40, -4, -4]$, but the correct calculation yields $[40, 1, -24]$. The above is the mathematically correct result. **Final answers:** $$-3A = \begin{bmatrix}24 & -15 & -12 \\ -21 & -12 & -15 \\ -15 & 18 & -12\end{bmatrix}$$ $$A - 3B = \begin{bmatrix}-8 & 2 & 16 \\ 22 & -11 & 2 \\ -7 & -21 & 28\end{bmatrix}$$ $$4A + 5B = \begin{bmatrix}-32 & 25 & -4 \\ 3 & 41 & 25 \\ 40 & 1 & -24\end{bmatrix}$$