A square matrix `P` satisfies `P^(2)=I-P` where `I` is identity matrix. If `P^(n)=5I-8P`, then `n` is

Explanation

To solve the problem, we start with the given equation involving the square matrix \( P \):

1. **Given**: \( P^2 = I - P \)

We can rearrange this to express \( P^2 \) in terms of \( P \) and \( I \).

2. **Step 1**: Find \( P^3 \)

To find \( P^3 \), we multiply both sides of the equation for \( P^2 \) by \( P \):
\[ P^3 = P \cdot P^2 = P(I - P) = P - P^2 \]
Now substitute \( P^2 \) from the original equation:
\[ P^3 = P - (I - P) = P - I + P = 2P - I \]

3. **Step 2**: Find \( P^4 \)

Next, we find \( P^4 \) by multiplying \( P^3 \) by \( P \):
\[ P^4 = P \cdot P^3 = P(2P - I) = 2P^2 - P \]
Substitute \( P^2 \) again:
\[ P^4 = 2(I - P) - P = 2I - 2P - P = 2I - 3P \]

4. **Step 3**: Find \( P^5 \)

Now, we find \( P^5 \):
\[ P^5 = P \cdot P^4 = P(2I - 3P) = 2P - 3P^2 \]
Substitute \( P^2 \):
\[ P^5 = 2P - 3(I - P) = 2P - 3I + 3P = 5P - 3I \]

5. **Step 4**: Find \( P^6 \)

Finally, we find \( P^6 \):
\[ P^6 = P \cdot P^5 = P(5P - 3I) = 5P^2 - 3P \]
Substitute \( P^2 \):
\[ P^6 = 5(I - P) - 3P = 5I - 5P - 3P = 5I - 8P \]

6. **Conclusion**: We have found that \( P^6 = 5I - 8P \).

Since we are given that \( P^n = 5I - 8P \), we can conclude that:
\[ n = 6 \]

Thus, the value of \( n \) is \( \boxed{6} \).