If A and B are two square matrices of order `3xx3` which satify `AB=A` and `BA=B`, then
Which of the following is true ?

Options

A. If matrix A is singular, then matrix B is nonsingular.
B. If matrix A is nonsingular, then materix B is singular.
C. If matrix A is singular, then matrix B is also singular.
D. Cannot say anything.

Explanation

To solve the problem, we need to analyze the given conditions for the two square matrices \( A \) and \( B \) of order \( 3 \times 3 \) that satisfy the equations \( AB = A \) and \( BA = B \).

### Step-by-Step Solution:

1. **Given Conditions**:
We start with the equations:
\[ AB = A \quad \text{(1)} \]
\[ BA = B \quad \text{(2)} \]

2. **Taking Determinants**:
We will take the determinant of both sides of equation (1):
\[ \text{det}(AB) = \text{det}(A) \]
Using the property of determinants that states \( \text{det}(AB) = \text{det}(A) \cdot \text{det}(B) \), we can rewrite this as:
\[ \text{det}(A) \cdot \text{det}(B) = \text{det}(A) \quad \text{(3)} \]

3. **Rearranging Equation (3)**:
Rearranging equation (3) gives us:
\[ \text{det}(A) \cdot \text{det}(B) - \text{det}(A) = 0 \]
Factoring out \( \text{det}(A) \):
\[ \text{det}(A) (\text{det}(B) - 1) = 0 \]

4. **Analyzing the Factors**:
From the factored equation, we have two cases:
- Case 1: \( \text{det}(A) = 0 \)
- Case 2: \( \text{det}(B) = 1 \)

5. **Taking Determinants of Equation (2)**:
Now, we take the determinant of both sides of equation (2):
\[ \text{det}(BA) = \text{det}(B) \]
Again, using the property of determinants:
\[ \text{det}(B) \cdot \text{det}(A) = \text{det}(B) \quad \text{(4)} \]

6. **Rearranging Equation (4)**:
Rearranging equation (4) gives us:
\[ \text{det}(B) \cdot \text{det}(A) - \text{det}(B) = 0 \]
Factoring out \( \text{det}(B) \):
\[ \text{det}(B) (\text{det}(A) - 1) = 0 \]

7. **Analyzing the Factors**:
From the factored equation, we have two cases:
- Case 1: \( \text{det}(B) = 0 \)
- Case 2: \( \text{det}(A) = 1 \)

8. **Conclusion**:
Combining the results from both equations:
- If \( \text{det}(A) = 0 \), then from (3) and (4), \( \text{det}(B) \) must also be \( 0 \).
- If \( \text{det}(B) = 0 \), then \( \text{det}(A) \) must also be \( 0 \).
- If \( \text{det}(A) = 1 \), then \( \text{det}(B) \) must be \( 0 \).
- If \( \text{det}(B) = 1 \), then \( \text{det}(A) \) must be \( 0 \).

Thus, we conclude that both matrices \( A \) and \( B \) are singular matrices.

### Final Answer:
Both matrices \( A \) and \( B \) are singular.