Question

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

Answer 1

\[Given: A = \left[ a_{ij} \right] \text{is a skew symmetric matrix} . \] 

\[ \Rightarrow a_{ij} = - a_{ji} \left[ \text{For all values of i}, j \right]\] 

\[ \Rightarrow a_{ii} = - a_{ii} \left[ \text{For all values of i} \right]\] 

\[ \Rightarrow a_{ij} = 0\] 

\[Now, \] 

\[ \sum^{}_i \sum^{}_j a_{ij} = a_{11} + a_{12} + a_{13} + . . . + a_{21} + a_{22} + a_{23} + . . . + a_{31} + a_{32} + a_{33 + . . .} \] 

\[ = 0 + a_{12} + a_{13} + . . . - a_{12} + 0 + a_{23} + . . . - a_{13} - a_{23} + 0 + . . . \] 

\[ = 0\] 

\[Thus, \] 

\[ \sum^{}_i \sum^{}_j a_{ij} = 0\]