Question

Let $A=\begin{pmatrix}1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5\end{pmatrix}$ and $P=\frac{1}{2}(A + A^T)$. Then:

• A. $P^T = P$
• B. $P^T = -P$
• C. $P^T = 2P$
• D. $P^T = -2P$
• E. $P^T = 3P$

Hint:

$\frac{1}{2}(A+A^T)$ is the symmetric part of matrix $A$, while $\frac{1}{2}(A-A^T)$ is the skew-symmetric part.

Answer 1

The Correct Option is A

Step 1: Concept
For any square matrix $A$, $(A + A^T)$ is always a symmetric matrix.

Step 2: Analysis
$P = \frac{1}{2}(A + A^T)$. Taking transpose on both sides: $P^T = (\frac{1}{2}(A + A^T))^T = \frac{1}{2}(A^T + (A^T)^T)$.

Step 3: Conclusion
$P^T = \frac{1}{2}(A^T + A) = P$. Since $P^T = P$, it is a symmetric matrix. Final Answer: (A)