Question

If A = [a_ij] is a skew-symmetric matrix, then write the value of $\sum _i$ a_ij.

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[a_{ij}\right]\mathrm{is}\mathrm{a}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrix}. \\ \Rightarrow a_{ij}=-a_{ij}\left[\mathrm{For}\mathrm{all}\mathrm{values}\mathrm{of}i,j\right] \\ \Rightarrow a_{ii}=-a_{ii}\left[\mathrm{For}\mathrm{all}\mathrm{values}\mathrm{of}i\right] \\ \Rightarrow a_{ij}+a_{ii}=0 \\ \Rightarrow 2a_{ii}=0 \\ \Rightarrow a_{ii}=0\left[\mathrm{For}\mathrm{all}\mathrm{values}\mathrm{of}i\right] \\ \sum _i{a}_{ii}=0+0+...+0\left[itimes\right] \\ =0 \\ \mathrm{Thus}, \\ \sum _i{a}_{ii}=0 \end{gathered}$$