1. Problem: Verify the associative law for matrix addition for matrices A, B, and C. 2. The associative law for matrix addition states that for any matrices A, B, and C of the same size: $$ (A + B) + C = A + (B + C) $$ 3. Given matrices: $$ A = \begin{bmatrix}3 & -1 \\ 2 & 4\end{bmatrix}, B = \begin{bmatrix}0 & 2 \\ 1 & -4\end{bmatrix}, C = \begin{bmatrix}4 & 1 \\ -3 & -2\end{bmatrix} $$ 4. Compute $(A + B) + C$: $$ A + B = \begin{bmatrix}3+0 & -1+2 \\ 2+1 & 4+(-4)\end{bmatrix} = \begin{bmatrix}3 & 1 \\ 3 & 0\end{bmatrix} $$ Then, $$ (A + B) + C = \begin{bmatrix}3+4 & 1+1 \\ 3+(-3) & 0+(-2)\end{bmatrix} = \begin{bmatrix}7 & 2 \\ 0 & -2\end{bmatrix} $$ 5. Compute $A + (B + C)$: $$ B + C = \begin{bmatrix}0+4 & 2+1 \\ 1+(-3) & -4+(-2)\end{bmatrix} = \begin{bmatrix}4 & 3 \\ -2 & -6\end{bmatrix} $$ Then, $$ A + (B + C) = \begin{bmatrix}3+4 & -1+3 \\ 2+(-2) & 4+(-6)\end{bmatrix} = \begin{bmatrix}7 & 2 \\ 0 & -2\end{bmatrix} $$ 6. Since $(A + B) + C = A + (B + C)$, the associative law for matrix addition holds. Final answer: $$ (A + B) + C = A + (B + C) = \begin{bmatrix}7 & 2 \\ 0 & -2\end{bmatrix} $$