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: Definition of a transpose. 
For a square matrix \( A \), the transpose \( A' \) is obtained by interchanging the rows and columns of \( A \). 
Step 2: Subtracting \( A' \) from \( A \). 
The matrix \( A - A' \) satisfies the property: \[ (A - A')' = A' - A = -(A - A'). \] Thus, \( A - A' \) is equal to the negative of its transpose, which is the definition of a skew symmetric matrix. 
Step 3: Conclusion. 
For any square matrix \( A \), \( A - A' \) is always a skew symmetric matrix. {10pt}