UPSEE Solved Model Paper 5

Let A = $[a_{ij}]$ be a skew - symmetric matrix Then $a_{ij}$ = - $a_{ji}$ for all i , j , ⇒ $a_{ij}$ = - $a_{ij}$ for all values of i = j ⇒ $2a_{ii}$ = 0 ⇒ $a_{ii}$ 0 for all i Now, let A be any square matrix then $\frac{1}{2}(A+A')$ = $\frac{1}{2}[A'+(A')]$ [Since (A + B) = A' + B' ] = $\frac{1}{2}$ (A' + A) [since (A')' = A] ⇒ $\frac{1}{2}$ (A + A') is symmetric matrix Also $\frac{1}{2}$ (A - A')' = $\frac{1}{2}$ [A' - (A')'] = $\frac{1}{2}$ (A' - A) = $-\frac{1}{2}$ (A - A') ⇒ $\frac{1}{2}$ (A - A') is skew - symmetric matrix. Hence option [a] is correct