If A is a square matrix, then A is skew symmetric if

Options

A. `A^(2)=A`
B. `A^(2)=l`
C. `A^(T)=A`
D. `A^(T)=-A`

Answer

To determine the condition under which a square matrix \( A \) is skew symmetric, we can follow these steps:

### Step-by-Step Solution:

1. **Definition of Skew Symmetric Matrix**:
A matrix \( A \) is said to be skew symmetric if it satisfies the condition:
\[ A^T = -A \]
where \( A^T \) is the transpose of matrix \( A \).

2. **Understanding the Transpose**:
The transpose of a matrix is obtained by flipping it over its diagonal. This means that the element at position \( (i, j) \) in matrix \( A \) becomes the element at position \( (j, i) \) in matrix \( A^T \).

3. **Condition for Skew Symmetry**:
For \( A \) to be skew symmetric, the above condition \( A^T = -A \) must hold true for all elements of the matrix. This implies that:
\[ a_{ij} = -a_{ji} \quad \text{for all } i, j \]
where \( a_{ij} \) represents the element in the \( i^{th} \) row and \( j^{th} \) column of matrix \( A \).

4. **Diagonal Elements**:
Since the diagonal elements are of the form \( a_{ii} = -a_{ii} \), it follows that:
\[ a_{ii} = 0 \quad \text{for all } i \]
This means that all diagonal elements of a skew symmetric matrix must be zero.

5. **Conclusion**:
Therefore, the condition for a square matrix \( A \) to be skew symmetric is:
\[ A^T = -A \]
or equivalently, \( a_{ij} = -a_{ji} \) for all \( i, j \) and \( a_{ii} = 0 \).

### Final Answer:
A square matrix \( A \) is skew symmetric if:
\[ A^T = -A \]