If for a `2xx2` matrix `A,A^(2)+I=O`, where I is identity matrix then A equals

Options

A. `[(1,0),(0,1)]`
B. `[(-i,0),(0,-i)]`
C. `[(1,2),(-1,1)]`
D. `[(-1,0),(0,-1)]`

Answer

To solve the problem, we need to find the matrix \( A \) such that \( A^2 + I = O \), where \( I \) is the identity matrix and \( O \) is the zero matrix.

### Step-by-step Solution:

1. **Understanding the Equation**:
We start with the equation:
\[ A^2 + I = O \]
This can be rewritten as:
\[ A^2 = -I \]
This means that the square of the matrix \( A \) is equal to the negative of the identity matrix.

2. **Identifying the Identity Matrix**:
For a \( 2 \times 2 \) matrix, the identity matrix \( I \) is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

3. **Finding Possible Matrices**:
We need to check various \( 2 \times 2 \) matrices to see if their square equals \( -I \).

4. **Option 1: \( A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)**:
\[ A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
\[ A^2 + I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \neq O \]
**Conclusion**: Option 1 is incorrect.

5. **Option 2: \( A = -I \)**:
\[ A = -\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]
\[ A^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I \]
\[ A^2 + I = I + I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \neq O \]
**Conclusion**: Option 2 is incorrect.

6. **Option 3: \( A = \begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix} \)**:
\[ A^2 = \begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 1 + (-1) \cdot 2 & 1 \cdot (-1) + (-1) \cdot 1 \\ 2 \cdot 1 + 1 \cdot 2 & 2 \cdot (-1) + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} -1 & -2 \\ 4 & -1 \end{pmatrix} \]
\[ A^2 + I = \begin{pmatrix} -1 & -2 \\ 4 & -1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 4 & 0 \end{pmatrix} \neq O \]
**Conclusion**: Option 3 is incorrect.

7. **Option 4: \( A = \begin{pmatrix} -1 & 0 \\ -1 & 0 \end{pmatrix} \)**:
\[ A^2 = \begin{pmatrix} -1 & 0 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} -1 & 0 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \]
\[ A^2 + I = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 1 & 1 \end{pmatrix} \neq O \]
**Conclusion**: Option 4 is incorrect.

### Final Conclusion:
None of the options provided satisfy the equation \( A^2 + I = O \).