Which of the following statement('s) is/are TRUE?

(A) Skew symmetric matrix of even order is always symmetric

(B) Skew symmetric matrix of odd order is non-singular

(C) Skew symmetric matrix of odd order is singular

(D) Skew symmetric matrix is always square matrix

Choose the correct answer from the options given below:

A matrix is skew-symmetric if $A^T = -A$.

This means all diagonal elements must be 0 (since $a_{ii} = -a_{ii}$ only if $a_{ii} = 0$).

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Statement (A): Skew symmetric matrix of even order is always symmetric

Skew-symmetric requires: $A^T = -A$

Symmetric requires: $A^T = A$

These are opposite conditions. A matrix cannot satisfy both unless it is the zero matrix.

For a $2 \times 2$ skew-symmetric matrix:

$A = \begin{bmatrix} 0 & 2 \\ -2 & 0 \end{bmatrix}$

$A^T = \begin{bmatrix} 0 & -2 \\ 2 & 0 \end{bmatrix} = -A$

Since $A \neq A^T$, the matrix is not symmetric.

Statement (A) is FALSE.

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Statement (C): Skew symmetric matrix of odd order is singular

For any skew-symmetric matrix $A$:

$\det(A) = \det(A^T)$

$\det(A^T) = \det(-A)$

$\det(-A) = (-1)^n \times \det(A)$ where $n$ is the order

For odd order ($n = 1, 3, 5, ...$):

$\det(A) = (-1)^n \times \det(A)$

$\det(A) = -\det(A)$

$2\det(A) = 0$

$\det(A) = 0$

The matrix is singular.

Statement (C) is TRUE.

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Statement (B): Skew symmetric matrix of odd order is non-singular

This is the opposite of statement (C).

Statement (B) is FALSE.

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Statement (D): Skew symmetric matrix is always square matrix

For the condition $A^T = -A$ to be valid:

If $A$ is $m \times n$, then $A^T$ is $n \times m$.

For $A^T = -A$, both must have the same dimensions.

This requires $m \times n = n \times m$, so $m = n$.

The matrix must be square.

Statement (D) is TRUE.

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Statements (C) and (D) are TRUE.