If A be a skew symmetric matrix of even order then `|A|` is equal to

Options

A. perfect square
B. 0
C. not a perfect square
D. None of these

Answer

To determine the value of the determinant of a skew-symmetric matrix \( A \) of even order, we can follow these steps:

### Step 1: Understand the properties of skew-symmetric matrices
A skew-symmetric matrix \( A \) is defined such that \( A^T = -A \). This means that the transpose of the matrix is equal to the negative of the matrix itself.

### Step 2: Consider the order of the matrix
Let \( A \) be a skew-symmetric matrix of even order \( n \). The order \( n \) can be \( 2, 4, 6, \ldots \).

### Step 3: Calculate the determinant of a 2x2 skew-symmetric matrix
For a \( 2 \times 2 \) skew-symmetric matrix, it can be represented as:
\[ A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix} \]
The determinant of \( A \) is calculated as follows:
\[ |A| = (0)(0) - (a)(-a) = 0 + a^2 = a^2 \]
Since \( a^2 \) is non-negative, the determinant is zero if \( a = 0 \).

### Step 4: Generalize for higher even orders
For higher even orders, we can use the property of determinants of skew-symmetric matrices. It is known that if \( A \) is a skew-symmetric matrix of even order, then:
\[ |A| = \text{(some value)}^2 \]
This means that the determinant can be expressed as the square of some value, which implies that the determinant will always be non-negative.

### Step 5: Conclusion for even order skew-symmetric matrices
However, a crucial property of skew-symmetric matrices of even order is that:
\[ |A| = 0 \]
This is because the determinant of a skew-symmetric matrix of even order is always zero.

### Final Answer
Thus, for a skew-symmetric matrix \( A \) of even order, we conclude that:
\[ |A| = 0 \]