Trace of a skew symmetric matrix is always equal to

Options

A. `sum a_(ij)`
B. ` sum a_(ii)`
C. zero
D. none of these

Answer

To find the trace of a skew-symmetric matrix, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Skew-Symmetric Matrix**:
A matrix \( A \) is called skew-symmetric if \( A^T = -A \). This means that the transpose of the matrix is equal to the negative of the matrix itself.

2. **Consider a 3x3 Skew-Symmetric Matrix**:
Let's denote a general 3x3 skew-symmetric matrix \( A \) as:
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
According to the property of skew-symmetric matrices, we have:
\[ A^T = \begin{pmatrix} a & d & g \\ b & e & h \\ c & f & i \end{pmatrix} = -A = \begin{pmatrix} -a & -b & -c \\ -d & -e & -f \\ -g & -h & -i \end{pmatrix} \]

3. **Equating Elements**:
From the equality \( A^T = -A \), we can equate the corresponding elements:
- \( a = -a \)
- \( e = -e \)
- \( i = -i \)
- \( b = -d \)
- \( c = -g \)
- \( f = -h \)

4. **Solving the Equations**:
From \( a = -a \), we can conclude that:
\[ 2a = 0 \implies a = 0 \]
Similarly, from \( e = -e \) and \( i = -i \), we get:
\[ 2e = 0 \implies e = 0 \]
\[ 2i = 0 \implies i = 0 \]

5. **Finding the Trace**:
The trace of a matrix is defined as the sum of its diagonal elements. Therefore, the trace of matrix \( A \) is:
\[ \text{Trace}(A) = a + e + i = 0 + 0 + 0 = 0 \]

6. **Conclusion**:
Thus, the trace of a skew-symmetric matrix is always equal to zero.

### Final Answer:
The trace of a skew-symmetric matrix is always equal to **0**.