A square matrix M of order 3 satisfies `M^(2)=I-M`, where `I` is an identity matrix of order 3. If `M^(n)=5I-8M`, then `n` is equal to _______.

Explanation

To solve the problem, we need to find the value of \( n \) such that \( M^n = 5I - 8M \) given that \( M^2 = I - M \).

### Step-by-step Solution:

1. **Start with the given equation**:
\[ M^2 = I - M \]

2. **Calculate \( M^3 \)**:
\[ M^3 = M^2 \cdot M = (I - M) \cdot M = M - M^2 \]
Now substitute \( M^2 \) from the first equation:
\[ M^3 = M - (I - M) = M - I + M = 2M - I \]

3. **Calculate \( M^4 \)**:
\[ M^4 = M^3 \cdot M = (2M - I) \cdot M = 2M^2 - M \]
Substitute \( M^2 \):
\[ M^4 = 2(I - M) - M = 2I - 2M - M = 2I - 3M \]

4. **Calculate \( M^5 \)**:
\[ M^5 = M^4 \cdot M = (2I - 3M) \cdot M = 2M - 3M^2 \]
Substitute \( M^2 \):
\[ M^5 = 2M - 3(I - M) = 2M - 3I + 3M = 5M - 3I \]

5. **Calculate \( M^6 \)**:
\[ M^6 = M^5 \cdot M = (5M - 3I) \cdot M = 5M^2 - 3M \]
Substitute \( M^2 \):
\[ M^6 = 5(I - M) - 3M = 5I - 5M - 3M = 5I - 8M \]

6. **Conclusion**:
We have found that:
\[ M^6 = 5I - 8M \]
Therefore, the value of \( n \) is:
\[ \boxed{6} \]