Verify that (A + B)^T = A^T + B^T

Given:

\( A = \begin{bmatrix} 2 & -3 \\ -7 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 2 & -4 \end{bmatrix} \)

To Verify:

\( (A + B)^T = A^T + B^T \)

Step 1: Find A + B

\[ A + B = \begin{bmatrix} 2+1 & -3+0 \\ -7+2 & 5-4 \end{bmatrix} = \begin{bmatrix} 3 & -3 \\ -5 & 1 \end{bmatrix} \]

Step 2: Find (A + B)^T

\[ (A + B)^T = \begin{bmatrix} 3 & -5 \\ -3 & 1 \end{bmatrix} \]

Step 3: Find A^T and B^T

\[ A^T = \begin{bmatrix} 2 & -7 \\ -3 & 5 \end{bmatrix}, \quad B^T = \begin{bmatrix} 1 & 2 \\ 0 & -4 \end{bmatrix} \]

Step 4: Find A^T + B^T

\[ A^T + B^T = \begin{bmatrix} 2+1 & -7+2 \\ -3+0 & 5-4 \end{bmatrix} = \begin{bmatrix} 3 & -5 \\ -3 & 1 \end{bmatrix} \]

Conclusion:

\[ (A + B)^T = A^T + B^T \]

Hence Verified.