Let A and B are two non - singular matrices such that `AB=BA^(2),B^(4)=I and A^(k)=I`, then k can be equal to

Options

A. 5
B. 10
C. 15
D. 16

Answer

To solve the problem, we have to find the value of \( k \) given the conditions on matrices \( A \) and \( B \). Let's go through the solution step by step.

### Step 1: Understand the Given Conditions
We have two non-singular matrices \( A \) and \( B \) such that:
1. \( AB = BA^2 \)
2. \( B^4 = I \) (where \( I \) is the identity matrix)
3. \( A^k = I \)

### Step 2: Analyze the Equation \( AB = BA^2 \)
From the equation \( AB = BA^2 \), we can manipulate it to express \( B \) in terms of \( A \):
\[ AB = BA^2 \implies A^{-1}AB = A^{-1}BA^2 \implies B = A^{-1}BA^2 \]
This shows that \( B \) can be expressed in terms of \( A \).

### Step 3: Use the Condition \( B^4 = I \)
Since \( B^4 = I \), this means that \( B \) is of finite order. The order of \( B \) is 4, meaning that \( B \) raised to the power of 4 gives the identity matrix.

### Step 4: Substitute \( B \) into the Equation
Now, we can use the equation \( AB = BA^2 \) repeatedly. Let's square both sides:
\[ (AB)^2 = (BA^2)(BA^2) \implies A(BA^2)B = B(BA^2)A^2 \]
This leads to:
\[ A^2B^2 = B^2A^4 \]

### Step 5: Simplify Using \( B^4 = I \)
Since \( B^4 = I \), we can express \( B^2 \) as:
\[ B^2 = B^{-2} \]
Thus, we can rewrite the equation:
\[ A^2B^2 = B^2A^4 \]
This implies that \( A^2 = A^4 \).

### Step 6: Solve for \( k \)
From \( A^2 = A^4 \), we can factor out \( A^2 \):
\[ A^2(A^2 - I) = 0 \]
Since \( A \) is non-singular, \( A^2 \neq 0 \), thus:
\[ A^2 - I = 0 \implies A^2 = I \]
This means that \( A \) has an order of 2, so \( A^2 = I \).

Now, since \( A^k = I \) and \( A^2 = I \), the possible values of \( k \) can be any even integer.

### Conclusion
The smallest positive integer \( k \) that satisfies \( A^k = I \) is \( k = 2 \). However, since we are looking for all possible values of \( k \), we can conclude that \( k \) can be any even integer.

Thus, the answer is:
\[ k = 2, 4, 6, \ldots \]