A and B are two matrices such that both AB and BA are defined
Assertion (A) : `(A + B) = A^(2) - B^(2)`
Reason (R) : `(A + B) ( A - B) = A^(2) - AB + BA - B^(2)`

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 provided regarding the matrices A and B.

### Step 1: Analyze the Assertion
The assertion states that:
\[ A + B = A^2 - B^2 \]
This is a claim that we need to verify.

### Step 2: Analyze the Reason
The reason states that:
\[ (A + B)(A - B) = A^2 - AB + BA - B^2 \]
This is a standard identity for the difference of squares, but we need to check if it holds given the matrices A and B.

### Step 3: Expand the Reason
Let's expand the left-hand side:
\[ (A + B)(A - B) = A^2 - AB + BA - B^2 \]
This is indeed a correct expansion based on the distributive property of matrices.

### Step 4: Compare the Assertion with the Reason
Now, we need to check if the assertion \( A + B = A^2 - B^2 \) can be derived from the reason.

From the reason, we have:
\[ A^2 - B^2 = (A + B)(A - B) + AB - BA \]
This shows that \( A^2 - B^2 \) is not simply \( A + B \) because it also includes the terms \( AB - BA \).

### Step 5: Conclusion
Since the assertion \( A + B = A^2 - B^2 \) cannot be derived from the reason, we conclude:
- The assertion (A) is **false**.
- The reason (R) is **true**.

### Final Answer
Thus, the correct option is that assertion A is false and reason R is true.

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