Question

If \( A = \left[\begin{array}{ccc} 83 & 74 & 41 \\ 93 & 96 & 31 \\ 24 & 15 & 79 \end{array} \right] \), then \(\text{det} (A - A^T) = \): 

• A. \( 0 \)
• B. \( -7851 \)
• C. \( 2442 \)
• D. \( 1 \)

Hint:

Remember that the determinant of any skew-symmetric matrix of odd order is always zero.

Answer 1

The Correct Option is A

We are given the matrix \( A = \left[\begin{array}{ccc} 83 & 74 & 41 \\ 93 & 96 & 31 \\ 24 & 15 & 79 \end{array} \right] \).

To find \( \text{det}(A - A^T) \), we first compute the transpose \( A^T \) of the matrix \( A \): 

\[ A^T = \left[\begin{array}{ccc} 83 & 93 & 24 \\ 74 & 96 & 15 \\ 41 & 31 & 79 \end{array} \right] \]

Now, subtract \( A^T \) from \( A \):

\[ A - A^T = \left[\begin{array}{ccc} 0 & -19 & 17 \\ 19 & 0 & -50 \\ -17 & 50 & 0 \end{array} \right] \]

The determinant of this matrix \( A - A^T \) is \( 0 \), since it is a skew-symmetric matrix (i.e., its transpose is equal to its negative), and the determinant of a skew-symmetric matrix of odd order is always \( 0 \).

Thus, \( \text{det}(A - A^T) = 0 \).