If for the matrix A, `A^(5)=I`, then `A^(-1)=`

Options

A. `A^(2)`
B. `A^(3)`
C. `A`
D. `A^(4)`

Answer

To solve the problem, we need to find the inverse of the matrix \( A \) given that \( A^5 = I \), where \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ A^5 = I \]

2. **Rewrite \( A^5 \)**:
We can express \( A^5 \) as:
\[ A^5 = A \cdot A^4 \]
Therefore, we have:
\[ A \cdot A^4 = I \]

3. **Multiply both sides by \( A^{-1} \)**:
To isolate \( A^4 \), we multiply both sides of the equation by \( A^{-1} \):
\[ A^{-1} \cdot (A \cdot A^4) = A^{-1} \cdot I \]

4. **Apply the property of inverses**:
Using the property that \( A^{-1} \cdot A = I \):
\[ I \cdot A^4 = A^{-1} \]
This simplifies to:
\[ A^4 = A^{-1} \]

5. **Conclusion**:
Thus, we find that:
\[ A^{-1} = A^4 \]

### Final Answer:
\[ A^{-1} = A^4 \]