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 2x2 matrix \( A \) given that \( A^2 = O \), where \( O \) is the null matrix.

### Step-by-Step Solution:

1. **Understanding the Matrix**:
Let \( A \) be a 2x2 matrix represented as:
\[ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
The trace of matrix \( A \), denoted as \( \text{tr}(A) \), is given by:
\[ \text{tr}(A) = a + d \]

2. **Using the Given Condition**:
We know that \( A^2 = O \). This means:
\[ A \cdot A = O \]
Calculating \( A^2 \):
\[ A^2 = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{pmatrix} \]
Setting this equal to the null matrix \( O \):
\[ \begin{pmatrix} a^2 + bc & ab + bd \\ ac + dc & bc + d^2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

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

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

Setting these equal gives:
\[ -a^2 = -d^2 \implies a^2 = d^2 \]
This implies \( a = d \) or \( a = -d \).

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

6. **Conclusion**:
Since \( A^2 = O \) implies that the eigenvalues of \( A \) are both zero, and the trace is the sum of the eigenvalues, we conclude that:
\[ \text{tr}(A) = 0 \]

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