1. The problem involves addition and scalar multiplication of matrices and vectors. 2. The general rules are: - Matrix addition: Add corresponding elements. - Scalar multiplication: Multiply each element by the scalar. 3. Given matrices/vectors: $$C = \begin{bmatrix}5 & 2 & 1 \\ 2 & 4 & 0\end{bmatrix}^T, \quad D = \begin{bmatrix}4 & 5 & 2 \\ 1 & 0 & -1\end{bmatrix}^T,$$ $$A = \begin{bmatrix}0 & 6 & 1 \\ 2 & 5 & 0 \\ 4 & 5 & -3\end{bmatrix}^T, \quad B = \begin{bmatrix}0 & 5 & -2 \\ 5 & 3 & 4 \\ 2 & 4 & -2\end{bmatrix}^T$$ 4. First, transpose each matrix to get their actual forms: $$C = \begin{bmatrix}5 & 2 \\ 1 & 2 \\ 4 & 0\end{bmatrix}, \quad D = \begin{bmatrix}4 & 1 \\ 5 & 0 \\ 2 & -1\end{bmatrix}$$ $$A = \begin{bmatrix}0 & 2 & 4 \\ 6 & 5 & 5 \\ 1 & 0 & -3\end{bmatrix}, \quad B = \begin{bmatrix}0 & 5 & 2 \\ 5 & 3 & 4 \\ -2 & 4 & -2\end{bmatrix}$$ 5. Example operation: Compute $C + D$ by adding corresponding elements: $$C + D = \begin{bmatrix}5+4 & 2+1 \\ 1+5 & 2+0 \\ 4+2 & 0+(-1)\end{bmatrix} = \begin{bmatrix}9 & 3 \\ 6 & 2 \\ 6 & -1\end{bmatrix}$$ 6. Example scalar multiplication: Multiply $A$ by scalar 2: $$2A = 2 \times \begin{bmatrix}0 & 2 & 4 \\ 6 & 5 & 5 \\ 1 & 0 & -3\end{bmatrix} = \begin{bmatrix}0 & 4 & 8 \\ 12 & 10 & 10 \\ 2 & 0 & -6\end{bmatrix}$$ This shows how to perform addition and scalar multiplication on the given matrices and vectors.