Question

If \( A = \begin{bmatrix} 2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2 \end{bmatrix} \) is expressed as a sum of a symmetric matrix \( P \) and a skew-symmetric matrix \( Q \), then \( P^T - Q^T \) equals:

• A. \( \begin{bmatrix} 8 & -16 & -4 \\ 2 & 8 & 7 \\ 6 & 14 & -16 \end{bmatrix} \)
• B. \( \begin{bmatrix} 2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2 \end{bmatrix} \)
• C. \( \begin{bmatrix} 2 & 4 & -5 \\ 0 & 3 & 7 \\ -3 & 1 & 2 \end{bmatrix} \)
• D. \( \begin{bmatrix} 1 & 0 & -\frac{3}{2} 2 & \frac{3}{2} & \frac{1}{2} -\frac{5}{2} & \frac{7}{2} & 1 \end{bmatrix} \)

Hint:

To decompose a matrix into symmetric and skew-symmetric parts: - Use \( \frac{1}{2}(A + A^T) \) for symmetric. - Use \( \frac{1}{2}(A - A^T) \) for skew-symmetric. - And remember: \( P^T - Q^T = P + Q = A \)

Answer 1

The Correct Option is B

Step 1: Understand the decomposition \( A = P + Q \)
Where:
\( P \) is symmetric \( (P = P^T) \)
\( Q \) is skew-symmetric \( (Q = -Q^T) \)
Using the standard identity for decomposing any square matrix \( A \) into symmetric and skew-symmetric parts: \[ P = \frac{1}{2}(A + A^T), \quad Q = \frac{1}{2}(A - A^T) \]
Step 2: Observe what's asked
We are asked to find \( P^T - Q^T \). But since \( P \) is symmetric and \( Q \) is skew-symmetric, we know: \[ P^T = P, \quad Q^T = -Q \Rightarrow P^T - Q^T = P + Q = A \] \[ \Rightarrow \boxed{P^T - Q^T = A} \] Hence, the correct answer is: \[ \begin{bmatrix} 2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2 \end{bmatrix} \]