Question

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

Answer 1

$$\begin{gathered} \mathrm{Given}:A=\left[a_{ij}\right]\mathrm{is}\mathrm{a}\mathrm{skew}\mathrm{symmetric}\mathrm{matrix}. \\ \Rightarrow a_{ij}=-a_{ji}\left[\mathrm{For}\mathrm{all}\mathrm{values}\mathrm{of}i\mathit{,}\mathit{}\mathit{}j\right] \\ \Rightarrow a_{ii}=-a_{ii}\left[\mathrm{For}\mathrm{all}\mathrm{values}\mathrm{of}i\right] \\ \Rightarrow a_{ij}=0 \\ \mathrm{Now}, \\ \sum _i^{}\sum _j^{}{a}_{ij}=a_{11}+a_{12}+a_{13}+...+a_{21}+a_{22}+a_{23}+...+a_{31}+a_{32}+a_{33+...} \\ =0+a_{12}+a_{13}+...-a_{12}+0+a_{23}+...-a_{13}-a_{23}+0+... \\ =0 \\ \mathrm{Thus}, \\ \sum _i^{}\sum _j^{}{a}_{ij}=0 \end{gathered}$$