If A and B arę square matrices of same order such that AB = A and BA = B, then

Options

A. `A^(2)=A,B^(2)=B`
B. `A^(2)=A,B^(2) neB`
C. `B^(2)=B,A^(2) neA`
D. `A^(2) ne A,B^(2) neB`

Answer

To solve the problem, we need to analyze the given conditions: \( AB = A \) and \( BA = B \).

### Step 1: Start with the equation \( AB = A \)

Given that \( AB = A \), we can rearrange this equation:

\[ AB - A = 0 \]

Factoring out \( A \) gives:

\[ A(B - I) = 0 \]

where \( I \) is the identity matrix.

### Step 2: Analyze the implication of \( A(B - I) = 0 \)

This equation implies that either \( A = 0 \) (the zero matrix) or \( B - I \) is not invertible. If \( B - I \) is not invertible, it means that \( B \) could be the identity matrix or a matrix that does not have a full rank.

### Step 3: Now consider the equation \( BA = B \)

Similarly, from \( BA = B \), we can rearrange this equation:

\[ BA - B = 0 \]

Factoring out \( B \) gives:

\[ B(A - I) = 0 \]

### Step 4: Analyze the implication of \( B(A - I) = 0 \)

This implies that either \( B = 0 \) (the zero matrix) or \( A - I \) is not invertible. If \( A - I \) is not invertible, it means that \( A \) could be the identity matrix or a matrix that does not have a full rank.

### Step 5: Conclude the implications

From the above steps, we can conclude that:

1. If \( A \) is not the zero matrix, then \( B \) must be the identity matrix.
2. If \( B \) is not the zero matrix, then \( A \) must be the identity matrix.

Thus, we can conclude that both \( A \) and \( B \) must be the identity matrix.

### Final Result

Therefore, we can conclude that:

\[ A = I \quad \text{and} \quad B = I \]