If A is a square matrix such that `A^(3) =I` then the value of `A^(-1) ` is equal to

Options

A. I
B. A
C. `A^(2) `
D. none of these

Answer

To solve the problem, we need to find the value of \( A^{-1} \) given that \( A^3 = I \), where \( I \) is the identity matrix.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given that \( A^3 = I \). This means that when the matrix \( A \) is multiplied by itself three times, the result is the identity matrix.

2. **Rewriting the Expression**:
We can express \( A^3 \) as:
\[ A^3 = A \times A \times A \]
Since \( A^3 = I \), we can write:
\[ A \times A \times A = I \]

3. **Finding the Inverse**:
We want to find \( A^{-1} \). From the property of inverses, we know that if \( A^3 = I \), we can manipulate this equation to find \( A^{-1} \).
We can rearrange \( A^3 = I \) as follows:
\[ A^3 \times A^{-1} = I \times A^{-1} \]
This simplifies to:
\[ A^2 = A^{-1} \]
Thus, we have:
\[ A^{-1} = A^2 \]

4. **Conclusion**:
Therefore, the value of \( A^{-1} \) is:
\[ A^{-1} = A^2 \]

### Final Answer:
\[ A^{-1} = A^2 \]