If P and Q are symmetric matrices and `PQ=QP` then which of the following is/are true (A) `QP^-1=P^-1Q` (B) PQ is skew symmetric (C) `P^-1Q` is symmetric (D) both (a) and (c)

Options

A. `QP^(-1) =P^(-1) Q `
B. ` PQ ` is skew symmetric
C. `P^(-1) Q ` is skew symmetric
D. Both (A) and (C )

Answer

To solve the problem, we need to analyze the properties of symmetric matrices and their products. Let's go through the steps systematically.

### Given:
- \( P \) and \( Q \) are symmetric matrices.
- \( PQ = QP \).

### To Prove:
We need to determine which of the following statements are true:
1. (A) \( QP^{-1} = P^{-1}Q \)
2. (B) \( PQ \) is skew symmetric.
3. (C) \( P^{-1}Q \) is symmetric.
4. (D) Both (A) and (C).

### Step-by-Step Solution:

**Step 1: Analyze Statement (A) \( QP^{-1} = P^{-1}Q \)**

Since \( PQ = QP \), we can take the inverse of both sides. The inverse of a product of matrices follows the rule \( (AB)^{-1} = B^{-1}A^{-1} \).

\[ (PQ)^{-1} = (QP)^{-1} \]

This gives us:

\[ Q^{-1}P^{-1} = P^{-1}Q^{-1} \]

Multiplying both sides by \( Q \) (assuming \( Q \) is invertible):

\[ QP^{-1} = P^{-1}Q \]

Thus, statement (A) is **true**.

**Step 2: Analyze Statement (B) \( PQ \) is skew symmetric.**

A matrix \( A \) is skew symmetric if \( A^T = -A \). Since \( P \) and \( Q \) are symmetric, we have:

\[ (PQ)^T = Q^T P^T = QP \]

Since \( PQ = QP \), we have:

\[ (PQ)^T = PQ \]

This means \( PQ \) is symmetric, not skew symmetric. Therefore, statement (B) is **false**.

**Step 3: Analyze Statement (C) \( P^{-1}Q \) is symmetric.**

To check if \( P^{-1}Q \) is symmetric, we compute its transpose:

\[ (P^{-1}Q)^T = Q^T (P^{-1})^T \]

Since \( P \) is symmetric, \( (P^{-1})^T = (P^T)^{-1} = P^{-1} \). Therefore:

\[ (P^{-1}Q)^T = QP^{-1} \]

From Step 1, we established that \( QP^{-1} = P^{-1}Q \). Thus:

\[ (P^{-1}Q)^T = P^{-1}Q \]

This shows that \( P^{-1}Q \) is symmetric. Therefore, statement (C) is **true**.

**Step 4: Conclusion**

Since statements (A) and (C) are true, and statement (B) is false, the correct answer is:

**(D) Both (A) and (C)**.

### Summary of Results:
- (A) True
- (B) False
- (C) True
- (D) True (since both (A) and (C) are true)