Question

If a square matrix \( A \) satisfies the condition \( A^T = -A \), then the matrix \( A \) is classified as a:

• A. Skew-symmetric matrix
• B. Symmetric matrix
• C. Identity matrix
• D. Orthogonal matrix

Hint:

To identify a skew-symmetric matrix instantly by eye, verify two traits: first, check if all entries along its main diagonal are exactly zero, and second, look to see if elements across from the diagonal mirror each other with opposite signs.

Answer 1

The Correct Option is A

Concept: Square matrices are classified into distinct structural groups based on how they behave when transformed by a transpose operation:

• Symmetric matrix: Transposing the array yields the exact same matrix layout (\( A^T = A \)).
• Skew-symmetric matrix: Transposing the array changes the sign of every element, yielding its negative (\( A^T = -A \)).

Step 1: Analyze the algebraic definition provided in the problem statement.
The problem states that transposing matrix \( A \) results in its negative: \[ A^T = -A \] This matches the exact definition of a skew-symmetric matrix.

Step 2: Review the structural properties of skew-symmetric elements.
For a matrix entry position, this means \( a_{ji} = -a_{ij} \). If we look at the main diagonal entries where column and row indices are equal (\( i = j \)): \[ a_{ii} = -a_{ii} \quad \Rightarrow \quad 2a_{ii} = 0 \quad \Rightarrow \quad a_{ii} = 0 \] This proves that every entry along the main diagonal of a skew-symmetric matrix must be exactly zero.