Step 1: Understanding the Concept:
A square matrix is defined as skew-symmetric if it is equal to the negative of its own transpose.
In other words, \(A^T = -A\).
This definition imposes strong restrictions on the elements of the matrix, particularly the diagonal elements, which must all be zero.
Furthermore, the determinant of such matrices follows a specific pattern based on the dimension (order) of the matrix.
The logic relies on how scalars interact with determinants and the fact that transposing a matrix does not change its determinant value.
Step 2: Key Formula or Approach:
We utilize two primary properties of determinants:
1. The determinant of a transpose is equal to the original determinant: \(|A^T| = |A|\).
2. Factoring out a scalar \(-1\) from an \(n \times n\) matrix: \(|kA| = k^n |A|\).
Step 3: Detailed Explanation:
From the definition of a skew-symmetric matrix:
\[ A^T = -A \]
Take the determinant of both sides of this equation:
\[ |A^T| = |-A| \]
Applying property 1 mentioned above:
\[ |A| = |-A| \]
The term \(-A\) can be viewed as the scalar \(-1\) multiplied by matrix \(A\).
Applying property 2, where \(n\) is the order of the matrix:
\[ |-A| = (-1)^n |A| \]
In this specific problem, the matrix order is \(5 \times 5\), so \(n = 5\):
\[ |A| = (-1)^5 |A| \]
Since \(5\) is an odd number, \((-1)^5 = -1\).
Substituting this back into the equation:
\[ |A| = -|A| \]
Bring all terms to one side:
\[ |A| + |A| = 0 \]
\[ 2|A| = 0 \]
Dividing by \(2\):
\[ |A| = 0 \]
This rigorous proof demonstrates that every skew-symmetric matrix with an odd order must have a determinant of zero.
Step 4: Final Answer:
The determinant of the given skew-symmetric matrix is \(0\), making (A) the correct answer.