Question

If $A$ is a skew-symmetric matrix of order $3$, then $|A|$ is:

• A. $1$
• B. $0$
• C. $-1$
• D. Depends on matrix

Hint:

The determinant of every skew-symmetric matrix of odd order is always zero.

Answer 1

The Correct Option is B

Concept: A skew-symmetric matrix satisfies: \[ A^T=-A \] An important property is: \[ |A^T|=|A| \] Also, \[ |-A|=(-1)^n|A| \] where $n$ is the order of the matrix.

Step 1: Use the skew-symmetric property.
Given: \[ A^T=-A \] Taking determinants: \[ |A^T|=|-A| \] Using determinant properties: \[ |A|=(-1)^n|A| \] Since order is $3$: \[ |A|=(-1)^3|A| \] \[ |A|=-|A| \]

Step 2: Simplify the equation.
Adding $|A|$ to both sides: \[ 2|A|=0 \] Therefore, \[ |A|=0 \] Hence, \[ \boxed{0} \]