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: Understanding the Concept:
We need to determine the relationship between a matrix P and its transpose \( P^T \). The matrix P is defined by a product involving another matrix A and its transpose \( A^T \). We need to check if P is symmetric (\( P^T=P \)), skew-symmetric (\( P^T=-P \)), or satisfies another relationship.
Step 2: Key Formula or Approach:
We will use the property of the transpose of a product of matrices: \( (XY)^T = Y^T X^T \).
We also use the property that the transpose of a transpose is the original matrix: \( (X^T)^T = X \).
We are assuming \( P = AA^T \). We will find \( P^T \) and compare it with P.
Step 3: Detailed Explanation:
Given \( P = AA^T \).
Let's find the transpose of P:
\[ P^T = (AA^T)^T \] Using the reverse order law for transposes \( (XY)^T = Y^T X^T \), where \( X=A \) and \( Y=A^T \):
\[ P^T = (A^T)^T A^T \] Now, using the property \( (X^T)^T = X \):
\[ P^T = A A^T \] By definition, we were given \( P = AA^T \).
Comparing the expressions, we see that:
\[ P^T = P \] This shows that P is a symmetric matrix. This property holds true for any matrix A. We do not need to use the specific values of A to prove this.
Step 4: Final Answer:
The relationship is \( P^T = P \).