If AB = A and BA = B, then

Options

A. `A^(2) = A`
B. `B^(2) = B`
C. `A^(2) = - A`
D. `B^(2) = - B `

Answer

To solve the problem where we have two matrices \( A \) and \( B \) such that \( AB = A \) and \( BA = B \), we will derive some properties of these matrices step by step.

### Step 1: Start with the given equations
We have two equations:
1. \( AB = A \)
2. \( BA = B \)

### Step 2: Rearranging the first equation
From the first equation \( AB = A \), we can rearrange it as:
\[ AB - A = 0 \]
Factoring out \( A \), we get:
\[ A(B - I) = 0 \]
where \( I \) is the identity matrix.

### Step 3: Analyzing the second equation
From the second equation \( BA = B \), we can rearrange it similarly:
\[ BA - B = 0 \]
Factoring out \( B \), we have:
\[ B(A - I) = 0 \]

### Step 4: Implications of the equations
The equations \( A(B - I) = 0 \) and \( B(A - I) = 0 \) imply that:
- Either \( A = 0 \) or \( B - I \) is not invertible.
- Either \( B = 0 \) or \( A - I \) is not invertible.

### Step 5: Exploring the case when \( A \) and \( B \) are non-zero
Assuming \( A \) and \( B \) are non-zero matrices, we can conclude that:
- \( B - I \) is singular, meaning \( B \) has an eigenvalue of 1.
- \( A - I \) is singular, meaning \( A \) has an eigenvalue of 1.

### Step 6: Conclusion
From the above deductions, we can conclude that:
- Both \( A \) and \( B \) are idempotent matrices, meaning \( A^2 = A \) and \( B^2 = B \).

### Final Result
Thus, we can summarize:
- \( A \) and \( B \) are idempotent matrices.

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