If for matrix A, `A^(3)=I` then `A^(-1)=`

Options

A. A
B. `A^(2)`
C. `A^(3)`
D. none of these

Answer

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

### Step-by-step Solution:

1. **Start with the given equation**:
\[ A^3 = I \]

2. **Multiply both sides by \( A^{-1} \)**:
\[ A^3 A^{-1} = I A^{-1} \]

3. **Simplify the left side**:
Since \( A^3 = A \cdot A \cdot A \), we can rewrite the left side:
\[ (A^2 A) A^{-1} = I A^{-1} \]
This simplifies to:
\[ A^2 (A A^{-1}) = A^{-1} \]

4. **Use the property of inverses**:
We know that \( A A^{-1} = I \), so we can substitute this into the equation:
\[ A^2 I = A^{-1} \]

5. **Simplify further**:
Since multiplying by the identity matrix does not change the matrix, we have:
\[ A^2 = A^{-1} \]

Thus, the inverse of matrix \( A \) is:
\[ A^{-1} = A^2 \]

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

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