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 Equation**:
We start with the equation:
\[ A^3 = I \]
This means that if we multiply the matrix \( A \) by itself three times, we get the identity matrix.

**Hint**: Recall that the identity matrix \( I \) has the property that \( AI = A \) for any matrix \( A \).

2. **Multiplying Both Sides by \( A^{-1} \)**:
To find \( A^{-1} \), we can manipulate the equation. We know that multiplying both sides of an equation by the same matrix (if it is invertible) does not change the equality. So, we multiply both sides of the equation \( A^3 = I \) by \( A^{-1} \):
\[ A^3 A^{-1} = I A^{-1} \]
This simplifies to:
\[ A^2 = A^{-1} \]

**Hint**: Remember that \( A A^{-1} = I \) is a key property of inverses.

3. **Conclusion**:
From the above manipulation, we have derived that:
\[ A^{-1} = A^2 \]

Thus, the value of \( A^{-1} \) is equal to \( A^2 \).

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