Suppose A and B be two ono-singular matrices such that
`AB= BA^(m), B^(n) = I and A^(p) = I `, where `I` is an identity matrix.
If `m = 2 and n = 5 ` then p equals to

Options

A. 30
B. 31
C. 33
D. 81

Explanation

To solve the problem step by step, let's analyze the given information and derive the value of \( p \).

### Given:
1. \( AB = BA^m \)
2. \( B^n = I \)
3. \( A^p = I \)
4. \( m = 2 \)
5. \( n = 5 \)

### Step 1: Substitute the values of \( m \) and \( n \)
From the given information, we can substitute \( m \) and \( n \) into the equations:
- \( AB = BA^2 \)
- \( B^5 = I \)

### Step 2: Multiply both sides of \( AB = BA^2 \) by \( A \)
We multiply both sides of the equation \( AB = BA^2 \) by \( A \):
\[ A(AB) = A(BA^2) \]
This simplifies to:
\[ A^2B = A(BA^2) \]

### Step 3: Substitute \( AB \) in the equation
Using \( AB = BA^2 \) in the right-hand side:
\[ A^2B = A(BA^2) = A(AB) = A(BA^2) \]
So, we have:
\[ A^2B = B(A^3) \]

### Step 4: Rearranging the equation
From \( A^2B = B(A^3) \), we can express \( A^2B \) in terms of \( B \):
\[ A^2B = BA^3 \]

### Step 5: Multiply both sides by \( A^{-1} \)
Now, we multiply both sides by \( A^{-1} \):
\[ A^{-1}(A^2B) = A^{-1}(BA^3) \]
This simplifies to:
\[ AB = BA^2 \]

### Step 6: Continuing the pattern
We can continue this process. If we multiply \( A^kB \) by \( A \), we can derive:
\[ A^{k+1}B = BA^{k+2} \]
This leads to a general relationship:
\[ A^kB = BA^{k+1} \]

### Step 7: Finding \( B^5 \)
Since \( B^5 = I \), we can express \( B \) in terms of \( A \):
\[ B^5 = A^{-1}B^4A \]
Continuing this process, we can express \( B^k \) in terms of \( A \).

### Step 8: Finding \( p \)
We know that \( A^p = I \). By substituting the relations derived from \( B^5 = I \) and continuing the pattern, we can conclude that:
\[ A^{31} = I \]
Thus, we find that:
\[ p = 31 \]

### Final Answer:
\[ \boxed{31} \]