If A and B are any two `2 xx 2` matrices such that AB=BA=B and B is not a zero matrix, what can you say about the matrix A ?

Answer

To solve the problem, we need to analyze the given conditions about the matrices A and B.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We have two matrices A and B, both of size \(2 \times 2\). We know that:
\[ AB = BA = B \]
and that B is not a zero matrix.

2. **Using the Identity Matrix Property**:
Recall that for any square matrix \(A\), multiplying it by the identity matrix \(I\) of the same order gives:
\[ A \cdot I = A \quad \text{and} \quad I \cdot A = A \]
where \(I\) is the identity matrix of order 2.

3. **Substituting the Identity Matrix**:
If we substitute \(A\) with the identity matrix \(I\), we can check if it satisfies the given conditions:
\[ AB = I \cdot B = B \quad \text{and} \quad BA = B \cdot I = B \]
This shows that if \(A\) is the identity matrix, both conditions \(AB = B\) and \(BA = B\) hold true.

4. **Conclusion**:
Since we have shown that substituting \(A\) with the identity matrix satisfies the conditions given in the problem, we can conclude:
\[ A = I \]
Hence, matrix \(A\) must be the identity matrix of order 2.

### Final Answer:
Thus, we can say that matrix \(A\) is the identity matrix of order 2.

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