Question

If $A_1,A_3{,......A}_{2n-1}$ are $n$ skew-symmetric matrices of same order, then $B=\sum _{r=1}^n(2r-1)({A_{2r-1})}^{2r-1}$ will be

• A. symmetric
• B. skew-symmetric
• C. neither symmetric nor skew-symmetric
• D. data not adequate

Answer 1

Answer: (B)

skew-symmetric

Given, $A_1,A_3{,......A}_{2n-1}$ are skew-symmetric matrices of same order.
$\Rightarrow A_1^T=-A_1$ ,$A_3^T=-A_3$, ..........$A_{2n-1}^T=-A_{2n-1}$ .....(1)
Now, $B={\sum }_{r=1}^n(2r-1)({A_{2r-1})}^{2r-1}$

$\Rightarrow B=A_1+3(A_3{)}^3+5(A_5{)}^5+.......+(2n-1)(A_{2n-1}{)}^{2n-1}$

$\Rightarrow B^T={A_1}^T+3(A_3^T{)}^3+.....+(2n-1\left)\right(A_{2n-1}^T{)}^{2n-1}$

$\Rightarrow B^T=-A_1-3(A_3{)}^3-.....-(2n-1\left)\right(A_{2n-1}{)}^{2n-1}$ (using (1))

$\Rightarrow B^T=-B$

Hence, B is skew-symmetric