If `A` and `P` are different matrices of order `n` satisfying `A^(3)=P^(3)` and `A^(2)P=P^(2)A` (where `|A-P| ne 0`) then `|A^(2)+P^(2)|` is equal to

Options

A. `n`
B. `0`
C. `|A||P|`
D. `|A+P|`

Explanation

To solve the problem, we need to find the value of \(|A^2 + P^2|\) given that \(A^3 = P^3\) and \(A^2P = P^2A\), with the condition that \(|A - P| \neq 0\).

### Step-by-step Solution:

1. **Start with the given equations**:
We have:
\[ A^3 = P^3 \]
and
\[ A^2P = P^2A. \]

2. **Rearranging the second equation**:
From \(A^2P = P^2A\), we can rearrange it to:
\[ A^2P - P^2A = 0. \]
This can be factored as:
\[ A^2P - P^2A = A^2P - AP^2 = 0 \implies A^2P - AP^2 = 0. \]

3. **Using the first equation**:
Since \(A^3 = P^3\), we can express this as:
\[ A^3 - P^3 = 0 \implies (A - P)(A^2 + AP + P^2) = 0. \]
Since \(|A - P| \neq 0\), we have:
\[ A^2 + AP + P^2 = 0. \]

4. **Express \(A^2 + P^2\)**:
From the previous step, we can express \(A^2 + P^2\) as:
\[ A^2 + P^2 = -AP. \]

5. **Determinant of the sum**:
We need to find \(|A^2 + P^2|\):
\[ |A^2 + P^2| = |-AP| = |A||P|. \]

6. **Final Result**:
Therefore, the value of \(|A^2 + P^2|\) is:
\[ |A^2 + P^2| = |A||P|. \]

### Conclusion:
The determinant \(|A^2 + P^2|\) is equal to \(|A||P|\).