STATEMENT -1 All positive odd integral powers of a skew - symmetric matrix are symmetric.
STATEMENT-2 : All positive even integral powers of a skew - symmetric matrix are symmetric.
STATEMENT-3 If A is a skew - symmetric matrix of even order then `|A|` is perfect square

Options

A. F T T
B. T T T
C. T F T
D. T T F

Answer

To solve the problem, we will analyze each statement regarding skew-symmetric matrices step by step.

### Step 1: Understanding Skew-Symmetric Matrices
A matrix \( A \) is called skew-symmetric if it satisfies the condition:
\[ A = -A^T \]
where \( A^T \) is the transpose of \( A \).

### Step 2: Evaluating Statement 1
**Statement 1:** All positive odd integral powers of a skew-symmetric matrix are symmetric.

To evaluate this, consider the odd power of a skew-symmetric matrix:
\[ A^1 = A \]
Taking the transpose:
\[ A^T = (-A) = -A \]
Since \( A^T \neq A \), the first statement is **false**.

### Step 3: Evaluating Statement 2
**Statement 2:** All positive even integral powers of a skew-symmetric matrix are symmetric.

For even powers, consider:
\[ A^2 = A \cdot A \]
Taking the transpose:
\[ (A^2)^T = (A \cdot A)^T = A^T \cdot A^T = (-A) \cdot (-A) = A^2 \]
Since \( (A^2)^T = A^2 \), it follows that \( A^2 \) is symmetric. Therefore, all positive even integral powers of a skew-symmetric matrix are symmetric, making this statement **true**.

### Step 4: Evaluating Statement 3
**Statement 3:** If \( A \) is a skew-symmetric matrix of even order, then \( |A| \) is a perfect square.

Consider a skew-symmetric matrix of even order (e.g., 2x2):
\[ A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix} \]
The determinant is calculated as:
\[ |A| = (0)(0) - (-a)(a) = a^2 \]
Since \( a^2 \) is a perfect square, this statement holds true for any skew-symmetric matrix of even order. Thus, this statement is **true**.

### Summary of Statements
- **Statement 1:** False
- **Statement 2:** True
- **Statement 3:** True