Question

If \[ A = \begin{bmatrix} 0 & -1 & 2 1 & 0 & 3 -2 & -3 & 0 \end{bmatrix} \] then \( A + 2A^T = \)

• A. \( -A^T \)
• B. \( A^T \)
• C. \( 2A^2 \)
• D. \( A \)

Hint:

The transpose of a matrix is found by swapping its rows and columns. Always remember to check the operation carefully.

Answer 1

The Correct Option is B

Step 1: Find the transpose of matrix \( A \).
The transpose of a matrix \( A \), denoted \( A^T \), is obtained by swapping its rows and columns. For matrix \( A \), we have:
\[ A^T = \begin{bmatrix} 0 & 1 & -2 -1 & 0 & -3 2 & 3 & 0 \end{bmatrix} \]

Step 2: Calculate \( A + 2A^T \).
Now, we calculate \( A + 2A^T \):
\[ A + 2A^T = \begin{bmatrix} 0 & -1 & 2 1 & 0 & 3 -2 & -3 & 0 \end{bmatrix} + 2 \times \begin{bmatrix} 0 & 1 & -2 -1 & 0 & -3 2 & 3 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -1 & 2 1 & 0 & 3 -2 & -3 & 0 \end{bmatrix} + \begin{bmatrix} 0 & 2 & -4 -2 & 0 & -6 4 & 6 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 & -2 -1 & 0 & -3 2 & 3 & 0 \end{bmatrix} \]
Thus, \( A + 2A^T = A^T \), and the correct answer is \( A^T \).