Let `A` and `B` be two non-singular square matrices such that `B ne I` and `AB^(2)=BA`. If `A^(3)-B^(-1)A^(3)B^(n)`, then value of `n` is

Options

A. `4`
B. `5`
C. `8`
D. `7`

Explanation

To solve the problem, we need to find the value of \( n \) such that the equation \( A^3 - B^{-1} A^3 B^n = 0 \) holds true, given that \( AB^2 = BA \).

### Step 1: Start with the given equation
We have the equation:
\[ A^3 - B^{-1} A^3 B^n = 0 \]
This implies:
\[ A^3 = B^{-1} A^3 B^n \]

### Step 2: Multiply both sides by \( B \)
To eliminate \( B^{-1} \) from the right side, we can multiply both sides by \( B \):
\[ B A^3 = A^3 B^n \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation:
\[ B A^3 - A^3 B^n = 0 \]
This can be factored as:
\[ A^3 B^n = B A^3 \]

### Step 4: Use the given condition \( AB^2 = BA \)
From the condition \( AB^2 = BA \), we can express \( A \) in terms of \( B \):
\[ A = B^{-1} AB^2 \]
However, we need to express \( A^3 \) in a useful form.

### Step 5: Analyze the implications of \( AB^2 = BA \)
Using the relation \( AB^2 = BA \), we can deduce:
\[ A B^2 = B A \implies A B = B A^{1/2} B \]
This suggests that \( A \) commutes with \( B \) under certain transformations.

### Step 6: Consider the implications for \( n \)
To satisfy \( A^3 B^n = B A^3 \), we can analyze the powers of \( B \). If we let \( n = 2 \), we can substitute back into the equation:
\[ A^3 B^2 = B A^3 \]
This holds true under the condition \( AB^2 = BA \).

### Step 7: Conclusion
Thus, the value of \( n \) that satisfies the equation is:
\[ \boxed{2} \]