If AB=A and BA=B then which of the following is/are true ?

Options

A. (a) A is idempotent
B. (b) B is idempotent
C. (c) `A^T` is idempotent
D. (d) none of these

Explanation

To solve the problem, we need to determine whether the matrices A and B are idempotent, and whether A transpose is idempotent based on the given conditions \( AB = A \) and \( BA = B \).

### Step-by-Step Solution:

1. **Understanding Idempotent Matrices**:
- A matrix \( M \) is said to be idempotent if \( M^2 = M \). This means that when the matrix is multiplied by itself, the result is the same as the original matrix.

2. **Given Conditions**:
- We are given two conditions:
- \( AB = A \) (1)
- \( BA = B \) (2)

3. **Checking if B is Idempotent**:
- Start with the equation \( BA = B \) (from condition (2)).
- Multiply both sides by B:
\[ B \cdot BA = B \cdot B \]
- This simplifies to:
\[ B^2 = B \]
- Since \( B^2 = B \), we conclude that matrix B is idempotent.

4. **Checking if A is Idempotent**:
- Now, use the equation \( AB = A \) (from condition (1)).
- Multiply both sides by A:
\[ A \cdot AB = A \cdot A \]
- This simplifies to:
\[ A^2 = A \]
- Since \( A^2 = A \), we conclude that matrix A is also idempotent.

5. **Checking if A Transpose is Idempotent**:
- We start with \( AB = A \) and take the transpose of both sides:
\[ (AB)^T = A^T \]
- Using the property of transposes, we can rewrite this as:
\[ B^T A^T = A^T \]
- Now, multiply both sides by \( A^T \):
\[ A^T B^T A^T = A^T A^T \]
- This simplifies to:
\[ A^T = A^T B^T \]
- From the above, we can conclude:
\[ (A^T)^2 = A^T \]
- Therefore, \( A^T \) is also idempotent.

### Conclusion:
- We have established that:
- A is idempotent.
- B is idempotent.
- A transpose is idempotent.

Thus, all the options are correct.