Let A and B are square matrices of order 3 such that `AB^(2)=BA and BA^(2)=AB`. If `(AB)^(3)=A^(3)B^(m)`, then m is equal to

Options

A. 3
B. 4
C. 5
D. 7

Answer

To solve the problem, we need to find the value of \( m \) given the conditions involving square matrices \( A \) and \( B \). Let's break down the solution step by step.

### Step 1: Understand the Given Conditions
We have two conditions:
1. \( AB^2 = BA \)
2. \( BA^2 = AB \)

### Step 2: Find \( (AB)^3 \)
We start by calculating \( (AB)^3 \):
\[ (AB)^3 = AB \cdot AB \cdot AB \]
This can be rewritten as:
\[ (AB)(AB)(AB) = (AB)(AB^2) = A(BA^2) \]

### Step 3: Substitute Using the Given Conditions
Using the second condition \( BA^2 = AB \), we can replace \( BA^2 \) in our expression:
\[ (AB)^3 = A(AB) = A^2B \]

### Step 4: Express \( (AB)^3 \) in Terms of \( A \) and \( B \)
Now we need to express \( (AB)^3 \) in terms of \( A^3 \) and \( B^m \). We can use the first condition \( AB^2 = BA \) to help us rearrange:
\[ (AB)^3 = A^2B^2B = A^2B^3 \]

### Step 5: Compare the Two Expressions
From the problem statement, we know:
\[ (AB)^3 = A^3B^m \]
Now we have:
\[ A^2B^3 = A^3B^m \]

### Step 6: Isolate \( B \)
To compare the coefficients of \( A \) and \( B \), we can factor out \( A^2 \):
\[ B^3 = A B^m \]

### Step 7: Compare the Powers of \( B \)
Since \( A \) is a matrix and does not affect the powers of \( B \), we can equate the powers of \( B \):
\[ 3 = m \]

### Conclusion
Thus, the value of \( m \) is:
\[ \boxed{3} \]