`A` and `B` are two square matrices such that `A^(2)B=BA` and if `(AB)^(10)=A^(k)B^(10)`, then `k` is

Options

A. `1001`
B. `1023`
C. `1042`
D. none of these

Explanation

To solve the problem, we need to find the value of \( k \) given the equations \( A^2 B = BA \) and \( (AB)^{10} = A^k B^{10} \).

### Step-by-Step Solution:

1. **Understanding the Given Equations**:
We have two square matrices \( A \) and \( B \) such that:
\[ A^2 B = BA \]
This implies that \( A \) and \( B \) commute under certain conditions.

2. **Finding \( AB^2 \)**:
We start by calculating \( AB^2 \):
\[ AB^2 = AB \cdot B = A(BA) = A(A^2B) = A^3B \]
Here, we used the relation \( A^2B = BA \).

3. **Finding \( AB^3 \)**:
Next, we calculate \( AB^3 \):
\[ AB^3 = AB^2 \cdot B = A^3B \cdot B = A^3(BA) = A^3(A^2B) = A^5B \]

4. **Generalizing the Pattern**:
From the calculations, we observe a pattern:
\[ AB^n = A^{2n - 1} B^n \]
We can prove this by induction or continue calculating for more values to confirm the pattern.

5. **Finding \( (AB)^{10} \)**:
Using the established pattern:
\[ (AB)^{10} = A^{2 \cdot 10 - 1} B^{10} = A^{19} B^{10} \]

6. **Equating to Given Expression**:
We know from the problem statement that:
\[ (AB)^{10} = A^k B^{10} \]
Therefore, we can set:
\[ A^{19} B^{10} = A^k B^{10} \]

7. **Finding \( k \)**:
By comparing the powers of \( A \) on both sides, we find:
\[ k = 19 \]

### Final Answer:
Thus, the value of \( k \) is \( 19 \).