Question

If the determinant of a 3rd order matrix \( A \) is \( K \), then the sum of the determinants of the matrices \( (AA^T) \) and \( (A - A^T) \) is:

• A. \( 2K \)
• B. \( 0 \)
• C. \( K^2 \)
• D. \( K \)

Hint:

For skew-symmetric matrices of odd order, the determinant is always zero. The determinant of \( AA^T \) is always the square of the determinant of \( A \).

Answer 1

The Correct Option is C

Step 1: Determinant Properties - For \( AA^T \): \[ \det(AA^T) = (\det A)^2 = K^2 \] - For \( A - A^T \): Since \( A - A^T \) is a skew-symmetric matrix of odd order, \[ \det(A - A^T) = 0 \] Step 2: Compute Sum of Determinants \[ \det(AA^T) + \det(A - A^T) = K^2 + 0 = K^2 \] Thus, the correct answer is \( K^2 \).