Assertion (A) : If A is a symmetric matrix , then B' AB is also symmetric
Reason (R) : (ABC)' = C'B'A'

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question step by step.

### Step 1: Understand the Assertion
The assertion states that if \( A \) is a symmetric matrix, then \( B'AB \) is also symmetric. A symmetric matrix \( A \) satisfies the property:
\[ A' = A \]

### Step 2: Analyze \( B'AB \)
To determine if \( B'AB \) is symmetric, we need to find the transpose of \( B'AB \):
\[ (B'AB)' = B'AB \]

### Step 3: Apply the Transpose Property
Using the property of transposes, we can write:
\[ (B'AB)' = B' (A) (B) = B' A' B \]
Since \( A \) is symmetric, we have \( A' = A \). Therefore:
\[ (B'AB)' = B' A B \]

### Step 4: Compare with the Original Expression
Now we need to check if \( (B'AB)' = B'AB \):
\[ B'AB = B'AB \]
This shows that \( B'AB \) is indeed symmetric.

### Conclusion on Assertion
Thus, the assertion \( A \) is true.

### Step 5: Understand the Reason
The reason states that \( (ABC)' = C'B'A' \). This is a property of transposes and is indeed true for any matrices \( A, B, \) and \( C \).

### Final Conclusion
Both the assertion \( A \) and the reason \( R \) are true, and the reason \( R \) correctly explains the assertion \( A \).

### Final Answer
Both assertion and reason are true, and the reason is the correct explanation of the assertion.

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