If A is a skew-symmetric matrix, then which of the following statements is NOT true?

(A) A is singular if order of A is odd

(B) A is non-singular

(C) $A^{2025}$ is a skew-symmetric matrix

(D) $A^{2025}$ is a symmetric matrix

(E) all diagonal elements of A are zeros

Choose the correct answer from the options given below:

A matrix A is skew-symmetric if $A^T = -A$ (transpose equals negative of the matrix).

For example: $A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{bmatrix}$

The elements across the diagonal are negatives of each other.

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For statement (A): A is singular if order of A is odd

For skew-symmetric matrices:

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

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

Since $A^T = -A$:

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

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

If $n$ is odd:

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

$2det(A) = 0$

$det(A) = 0$

The matrix is singular. Statement (A) is true.

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For statement (B): A is non-singular

From statement (A), odd-order skew-symmetric matrices are always singular since $det(A) = 0$.

Statement (B) is NOT true in general.

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For statement (C): $A^{2025}$ is skew-symmetric

For odd powers of skew-symmetric matrices:

$(A^k)^T = (A^T)^k$

$(A^k)^T = (-A)^k$

$(A^k)^T = (-1)^k \times A^k$

Since 2025 is odd:

$(A^{2025})^T = (-1)^{2025} \times A^{2025}$

$(A^{2025})^T = -A^{2025}$

This satisfies the definition of skew-symmetric. Statement (C) is true.

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For statement (D): $A^{2025}$ is symmetric

From statement (C), $A^{2025}$ is skew-symmetric.

A symmetric matrix requires $A^T = A$, but $(A^{2025})^T = -A^{2025}$.

Statement (D) is NOT true.

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For statement (E): All diagonal elements of A are zeros

For a skew-symmetric matrix: $a_{ij} = -a_{ji}$

For diagonal elements where $i = j$:

$a_{ii} = -a_{ii}$

$2a_{ii} = 0$

$a_{ii} = 0$

All diagonal elements must be zero. Statement (E) is true.

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Statements that are NOT true: (B) and (D)

The answer depends on which option corresponds to statements (B) and (D).