Statement -1 (Assertion) and Statement - 2 (Reason)
Each of these examples also has four alternative choices,
ONLY ONE of which is the correct answer. You have to
select the correct choice as given below
Statement-1 If A and B are two matrices such
that AB = B, BA = A, then ` A^(2) + B^(2) = A+B.`
Statement-2 A and B are idempotent motrices, then
`A^(2) = A, B^(2) = B`.

Options

A. Statement - 1 is true, Statement - 2 is true , Statement - 2
is correct explanaction for Statement -2
B. Statement -1 is true, Statement - 2 is true, Statement - 2
is not a correct explanation for Statement-2
C. Statement-1 is true, Statement-2 is false
D. Statement-1 is false, Statement-2 is ttrue

Explanation

To solve the problem, we need to analyze the two statements given and determine their validity.

### Step 1: Analyze Statement 1
**Statement 1:** If \( A \) and \( B \) are two matrices such that \( AB = B \) and \( BA = A \), then \( A^2 + B^2 = A + B \).

1. Start with the equation \( AB = B \).
2. Multiply both sides by \( B \):
\[ AB \cdot B = B \cdot B \implies A(BB) = B^2 \implies A = B^2 \quad \text{(since \( BB = B \))} \]
This gives us Equation (1): \( B = B^2 \).

3. Now, consider the second equation \( BA = A \).
4. Multiply both sides by \( A \):
\[ BA \cdot A = A \cdot A \implies B(AA) = A^2 \implies B = A^2 \quad \text{(since \( AA = A \))} \]
This gives us Equation (2): \( A = A^2 \).

5. Now, we have:
- From Equation (1): \( B^2 = B \)
- From Equation (2): \( A^2 = A \)

6. Adding these two equations:
\[ A^2 + B^2 = A + B \]
Thus, Statement 1 is **true**.

### Step 2: Analyze Statement 2
**Statement 2:** \( A \) and \( B \) are idempotent matrices, then \( A^2 = A \) and \( B^2 = B \).

1. By definition, a matrix \( A \) is idempotent if \( A^2 = A \).
2. Similarly, \( B \) is idempotent if \( B^2 = B \).
3. Therefore, Statement 2 is also **true**.

### Step 3: Determine the Relationship between Statements
- Both statements are true.
- However, Statement 2 does not provide a direct explanation for Statement 1. Statement 1 is a specific case derived from the properties of matrices \( A \) and \( B \), while Statement 2 is a general property of idempotent matrices.

### Conclusion
- Both statements are true, but Statement 2 is not the correct explanation for Statement 1. Therefore, the answer is option B.