Let A and B be square matrices of same order satisfying AB =A and BA =B, then `A^(2) B^(2) ` equals (O being zero matrices of the same order as B)

Options

A. A
B. B
C. I
D. O

Answer

To solve the problem, we need to find \( A^2 B^2 \) given the conditions \( AB = A \) and \( BA = B \).

### Step 1: Analyze the given equations
We start with the 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:
\[ AB - A = 0 \implies A(B - I) = 0 \]
This implies that either \( A = 0 \) (the zero matrix) or \( B - I \) is not invertible.

### Step 3: Rearranging the second equation
From the second equation \( BA = B \), we can rearrange it as:
\[ BA - B = 0 \implies B(A - I) = 0 \]
This implies that either \( B = 0 \) (the zero matrix) or \( A - I \) is not invertible.

### Step 4: Consider the implications of the equations
From the two rearranged equations, we have:
1. \( A(B - I) = 0 \)
2. \( B(A - I) = 0 \)

If \( A \) and \( B \) are not the zero matrices, then \( B - I \) and \( A - I \) must be singular (not invertible).

### Step 5: Multiply both sides of the equations
Now, we will multiply the two original equations:
\[ AB \cdot BA = A \cdot B \]
Substituting \( AB = A \) and \( BA = B \):
\[ A \cdot B = A \cdot B \]
This does not provide new information.

### Step 6: Calculate \( A^2 B^2 \)
Now, we calculate \( A^2 B^2 \):
\[ A^2 B^2 = A \cdot A \cdot B \cdot B \]
Using the original equations, we can express \( A^2 \) and \( B^2 \):
1. From \( AB = A \), we can say \( A = AB \).
2. From \( BA = B \), we can say \( B = BA \).

### Step 7: Substitute \( A \) and \( B \)
Now substituting \( A \) and \( B \):
\[ A^2 = A \cdot A = A \cdot (AB) = (AA)B = A^2B \]
And similarly for \( B^2 \):
\[ B^2 = B \cdot B = B \cdot (BA) = (BB)A = B^2A \]

### Step 8: Conclusion
Given that both \( A \) and \( B \) can be the identity matrix or the zero matrix, we conclude:
\[ A^2 B^2 = 0 \quad \text{(if either A or B is the zero matrix)} \]
Thus, the final result is:
\[ A^2 B^2 = O \]
where \( O \) is the zero matrix of the same order as \( A \) and \( B \).