Question

If A is a skew-symmetric matrix of order 3 × 3, then |A| = ______________.

Answer 1

Given: A is a skew-symmetric matrix of order 3 × 3

$$\begin{gathered} A=-A^T \\ \mathrm{Taking}\mathrm{determinant}\mathrm{on}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow \left|A\right|=\left|-A^T\right| \\ \Rightarrow \left|A\right|={\left(-1\right)}^3\left|A^T\right|\left(\because \mathrm{Order}\mathrm{of}A\mathrm{is}3\times 3\right) \\ \Rightarrow \left|A\right|={\left(-1\right)}^3\left|A\right|\left(\because \left|A^T\right|=\left|A\right|\right) \\ \Rightarrow \left|A\right|=-\left|A\right| \\ \Rightarrow \left|A\right|+\left|A\right|=0 \\ \Rightarrow 2\left|A\right|=0 \\ \Rightarrow \left|A\right|=0 \end{gathered}$$

Hence, |A| = 0.