Let a be a matrix of order `2xx2` such that `A^(2)=O`.
tr (A) is equal to

Options

A. `1`
B. `0`
C. `-1`
D. none of these

Explanation

To solve the problem, we need to find the trace of a matrix \( A \) of order \( 2 \times 2 \) given that \( A^2 = O \), where \( O \) is the null matrix.

### Step-by-Step Solution:

1. **Understanding the Matrix**:
Let \( A \) be a \( 2 \times 2 \) matrix represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

2. **Given Condition**:
We know that \( A^2 = O \). This means:
\[ A \cdot A = O \]

3. **Calculating \( A^2 \)**:
We compute \( A^2 \):
\[ A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{pmatrix} \]
Setting this equal to the null matrix \( O \):
\[ \begin{pmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

4. **Setting Up Equations**:
From the above equality, we derive the following equations:
- \( a^2 + bc = 0 \) (1)
- \( ab + bd = 0 \) (2)
- \( ac + dc = 0 \) (3)
- \( bc + d^2 = 0 \) (4)

5. **Finding the Trace**:
The trace of matrix \( A \) is given by:
\[ \text{tr}(A) = a + d \]

6. **Using the Equations**:
From equations (1) and (4), we can express \( d^2 \) and \( a^2 \):
- From (1): \( a^2 = -bc \)
- From (4): \( d^2 = -bc \)

This implies:
\[ a^2 = d^2 \]
Therefore, we can conclude that:
\[ a = d \quad \text{or} \quad a = -d \]

7. **Substituting into the Trace**:
If \( a = d \), then:
\[ \text{tr}(A) = a + a = 2a \]
If \( a = -d \), then:
\[ \text{tr}(A) = a - a = 0 \]

8. **Conclusion**:
Since \( A^2 = O \) implies that the eigenvalues of \( A \) are both zero, the trace must also be zero. Therefore:
\[ \text{tr}(A) = 0 \]

### Final Answer:
\[ \text{tr}(A) = 0 \]