Question

Statement I. Determine if a skew-symmetric matrix of order $3$ is zero.

Statement II. For any matrix $A,\left|A^{\top }\right|=\left|A\right|$ and $\left|-A\right|=-\left|A\right|$. Where $\left|B\right|$ denotes the determinant of matrix $B$. Then,

• A. Statement I is correct, Statement II is incorrect
• B. Both statements are correct
• C. Both statements are incorrect.
• D. Statement I is incorrect and Statement II is correct

Answer 1

The correct option is A

Statement I is correct, Statement II is incorrect

Explanation for the correct option:

Finding the value of $\left|A\right|$:

Let $A$be a skew-symmetric matrix so

$A^T=-A$

Now taking determinants on both the sides we get

$$\begin{gathered} \left|A^T\right|=\left|-A\right| \\ \Rightarrow \left|A\right|=\left(-1\right)\left|A\right|\left(\therefore \left|A\right|=\left|A^T\right|\right) \\ \Rightarrow \left|A\right|={\left(-1\right)}^n\left|A\right| \end{gathered}$$

Therefore,

$$\begin{gathered} \left|A\right|={\left(-1\right)}^3\left|A\right| \\ \Rightarrow \left|A\right|=-\left|A\right| \end{gathered}$$

Hence the statement II has been proven.

Since $n=3$ is odd

$$\begin{gathered} \Rightarrow \left|A\right|=-\left|A\right| \\ \Rightarrow \left|A\right|+\left|A\right|=0 \\ \Rightarrow 2\left|A\right|=0 \\ \Rightarrow \left|A\right|=0 \end{gathered}$$

Hence the statement I has been proven.

Therefore, the correct answer is option (A).