Which of the following is correct?

Options

A. Skew-symmetric matrix of even order is always singula
B. Skew-symmetric matrix of odd order is non-singular
C. Skew-symmetric matrix of odd order is singular
D. None of the above

Answer

To determine which of the given statements about skew-symmetric matrices is correct, we need to follow a systematic approach based on the properties of skew-symmetric matrices. Here’s a step-by-step solution:

### Step 1: Understand the Definition of Skew-Symmetric Matrix
A matrix \( A \) is called skew-symmetric if \( A^T = -A \), where \( A^T \) is the transpose of matrix \( A \).

**Hint:** Recall that for a skew-symmetric matrix, the elements satisfy \( a_{ij} = -a_{ji} \).

### Step 2: Determine the Determinant of a Skew-Symmetric Matrix
From the properties of determinants, we know that:
\[ \text{det}(A^T) = \text{det}(A) \]
For a skew-symmetric matrix, we have:
\[ \text{det}(A^T) = \text{det}(-A) \]

**Hint:** Remember that the determinant of a matrix multiplied by a scalar can be expressed as \( \text{det}(cA) = c^n \text{det}(A) \) for an \( n \times n \) matrix.

### Step 3: Relate the Determinants
Using the property of determinants:
\[ \text{det}(-A) = (-1)^n \text{det}(A) \]
where \( n \) is the order of the matrix. Thus, we can write:
\[ \text{det}(A) = \text{det}(-A) = (-1)^n \text{det}(A) \]

**Hint:** Consider what happens when \( n \) is odd versus when \( n \) is even.

### Step 4: Analyze the Case When \( n \) is Odd
If \( n \) is odd, then \( (-1)^n = -1 \). Therefore, we have:
\[ \text{det}(A) = -\text{det}(A) \]
This implies:
\[ 2 \cdot \text{det}(A) = 0 \quad \Rightarrow \quad \text{det}(A) = 0 \]

**Hint:** What does it mean when the determinant of a matrix is zero?

### Step 5: Conclusion
Since we have established that if \( A \) is a skew-symmetric matrix of odd order, then \( \text{det}(A) = 0 \). This means that \( A \) is a singular matrix. Therefore, the statement that "a skew-symmetric matrix of odd order is always singular" is correct.

**Final Answer:** The correct statement is option C: "A skew-symmetric matrix of odd order is always singular."