Let A = `((a,b),(c,d))` be such that `A^(3)=O` but `Ane O` then

Options

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

Answer

To solve the problem, we need to analyze the properties of the matrix \( A \) given that \( A^3 = O \) (the null matrix) and \( A \neq O \).

### Step-by-Step Solution:

1. **Given Information**: We know that \( A^3 = O \) and \( A \neq O \).

2. **Post-Multiplying by \( A^{-1} \)**: Since \( A \) is not the null matrix, we can consider the possibility of finding \( A^{-1} \) (the inverse of \( A \)). However, we need to be careful because \( A \) being nilpotent (since \( A^3 = O \)) implies that it cannot have an inverse. Thus, we cannot directly multiply by \( A^{-1} \).

3. **Analyzing \( A^3 = O \)**: Since \( A^3 = O \), we can express this as:
\[ A \cdot A \cdot A = O \]
This indicates that multiplying \( A \) by itself three times results in the null matrix.

4. **Exploring the Implications**: From the property of nilpotent matrices, we can conclude that:
- \( A^2 \) cannot be the identity matrix \( I \) because if \( A^2 = I \), then \( A^3 \) would not be \( O \).
- Therefore, \( A^2 \) must also be a nilpotent matrix.

5. **Finding \( A^2 \)**: Since \( A^3 = O \) implies that \( A^2 \) must also be such that:
\[ A^2 \cdot A = O \]
This means \( A^2 \) must also be a non-invertible matrix (since it leads to the null matrix when multiplied by \( A \)).

6. **Conclusion**: The only conclusion we can draw is that:
\[ A^2 = O \]
This is consistent with the properties of nilpotent matrices.

### Final Result:
Thus, we conclude that \( A^2 = O \).