1. **Stating the problem:** We are given matrices A, B, C, D, and E and asked to compute various matrix operations including sums, differences, scalar multiples, transposes, and traces. 2. **Recall the trace definition:** The trace of a square matrix is the sum of its diagonal elements. 3. **Compute tr(D):** Matrix D = \begin{bmatrix}1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4\end{bmatrix} The diagonal elements are 1, 0, and 4. So, $$\text{tr}(D) = 1 + 0 + 4 = 5$$ 4. **Compute tr(D - 3E):** First compute 3E: $$3E = 3 \times \begin{bmatrix}6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3\end{bmatrix} = \begin{bmatrix}18 & 3 & 9 \\ -3 & 3 & 6 \\ 12 & 3 & 9\end{bmatrix}$$ Now compute D - 3E: $$D - 3E = \begin{bmatrix}1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4\end{bmatrix} - \begin{bmatrix}18 & 3 & 9 \\ -3 & 3 & 6 \\ 12 & 3 & 9\end{bmatrix} = \begin{bmatrix}1-18 & 5-3 & 2-9 \\ -1+3 & 0-3 & 1-6 \\ 3-12 & 2-3 & 4-9\end{bmatrix} = \begin{bmatrix}-17 & 2 & -7 \\ 2 & -3 & -5 \\ -9 & -1 & -5\end{bmatrix}$$ The diagonal elements of D - 3E are -17, -3, and -5. So, $$\text{tr}(D - 3E) = -17 + (-3) + (-5) = -25$$ 5. **Compute 4 tr(7B):** First compute 7B: $$7B = 7 \times \begin{bmatrix}4 & -1 \\ 0 & 2\end{bmatrix} = \begin{bmatrix}28 & -7 \\ 0 & 14\end{bmatrix}$$ The diagonal elements of 7B are 28 and 14. So, $$\text{tr}(7B) = 28 + 14 = 42$$ Multiply by 4: $$4 \times 42 = 168$$ 6. **Compute tr(A):** Matrix A = \begin{bmatrix}3 & 0 \\ -1 & 2 \\ 1 & 1\end{bmatrix} Note: A is a 3x2 matrix, not square, so trace is not defined. Since trace is only defined for square matrices, tr(A) is undefined. **Final answers:** - k) $$4 \text{tr}(7B) = 168$$ - l) $$\text{tr}(A) \text{ is undefined (A is not square)}$$