Question

For any square matrix \( A \), \( (A - A') \) is always: {5pt}

• A. an identity matrix
• B. a null matrix
• C. a skew symmetric matrix
• D. a symmetric matrix
  {5pt}

Hint:

For square matrices: - \( A + A' \) is symmetric. - \( A - A' \) is skew symmetric.

Answer 1

The Correct Option is C

Step 1: Defining a Transpose.
The transpose of a square matrix \( A \), denoted as \( A' \), is formed by swapping its rows and columns.
Step 2: Calculating \( A - A' \).
The matrix \( A - A' \) has the property: \[ (A - A')' = A' - A = -(A - A'). \] This indicates that \( A - A' \) is the negative of its transpose, fulfilling the definition of a skew-symmetric matrix.
Step 3: Final Determination.
For any given square matrix \( A \), the expression \( A - A' \) consistently results in a skew-symmetric matrix. {10pt}