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.
Which of the following orderd triplet `(m, n, p)` is false?

Options

A. `(3, 2, 80)`
B. `(6, 3, 215)`
C. `(8, 3, 510)`
D. `(2, 8, 255)`

Explanation

To solve the problem, we need to analyze the given conditions involving the matrices \( A \) and \( B \). We have the following equations:

1. \( AB = BA^m \)
2. \( B^n = I \)
3. \( A^p = I \)

Where \( I \) is the identity matrix.

### Step 1: Analyze the first equation \( AB = BA^m \)

From the equation \( AB = BA^m \), we can rearrange it to express \( B \) in terms of \( A \):

\[ B = A^{-1}BA^m \]

This suggests that \( B \) can be transformed by multiplying it with \( A^{-1} \) and \( A^m \).

### Step 2: Use the second equation \( B^n = I \)

Given \( B^n = I \), we can express \( B \) in terms of its powers. Since \( B \) is an unsingular matrix, \( B^{-1} \) exists, and we can also express:

\[ B = B^{-1}B^n \]

This implies that \( B \) has a finite order \( n \).

### Step 3: Use the third equation \( A^p = I \)

Similarly, from \( A^p = I \), we know that \( A \) also has a finite order \( p \).

### Step 4: Relate \( p \), \( m \), and \( n \)

Using the first equation \( AB = BA^m \) repeatedly, we can derive relationships between \( m \), \( n \), and \( p \). We can express \( B^n \) in terms of \( A \):

\[ B^n = A^{-1}B A^{mn-1} \]

Continuing this process, we can derive that:

\[ B^n = A^{-1}B^{n-1}A^{m(n-1)} \]

This pattern suggests that we can express \( p \) in terms of \( m \) and \( n \):

\[ p = m^{n-1} \]

### Step 5: Check the given options

Now we will check the options provided to find which ordered triplet \( (m, n, p) \) is false.

1. **Option A**: \( (3, 2, 8) \)
- Check: \( p = m^{n-1} = 3^{2-1} = 3^1 = 3 \) (not equal to 8)

2. **Option B**: \( (6, 3, 215) \)
- Check: \( p = m^{n-1} = 6^{3-1} = 6^2 = 36 \) (not equal to 215)

3. **Option C**: \( (8, 3, 510) \)
- Check: \( p = m^{n-1} = 8^{3-1} = 8^2 = 64 \) (not equal to 510)

4. **Option D**: \( (2, 8, 255) \)
- Check: \( p = m^{n-1} = 2^{8-1} = 2^7 = 128 \) (not equal to 255)

### Conclusion

From the checks, we find that **Options A, B, C, and D** do not satisfy the equation \( p = m^{n-1} \). However, we need to find the false triplet. The ordered triplet that does not satisfy the equation is:

- **Option A**: \( (3, 2, 8) \) is false.