Question

If $A_1,A_3,.....,A_{2n-1}$ are $n$ skew symmetric matrices of same order $5\times 5$, then $B={\sum }_{r=1}^n(2r-1)(A_{2r-1}{)}^{2r-1}$ is a

• A. Symmetric matrix
• B. Singular matrix
• C. Non invertible matrix
• D. Skew-symmetric matrix

Answer 1

Answer: (B), (C), (D)

The correct options are
B Singular matrix
C Non invertible matrix
D Skew-symmetric matrix

$B=A_1+3A_3^3+5A_5^5+....+(2n-1)A_{2n-1}^{2n-1}$
$\Rightarrow B^T=A_1^T+3(A_3^T{)}^3+5(A_5^T{)}^5+....+(2n-1)(A_{2n-1}^T{)}^{2n-1}$
We know that, $A_1,A_3,.....,A_{2n-1}$ are $n$ skew symmetric matrices.
Therefore, $A_1^T=-A_1,A_3^T=-A_3,.......$
$\therefore B^T=-(A_1+3A_3^3+5A_5^5+....+(2n-1\left)A_{2n-1}^{2n-1}\right)B^T=-B$
Therefore, B is a skew-symmetric matrix of order $5\times 5$
Also, $|B^T|=|-B|\Rightarrow |B|=-|B|\Rightarrow |B|=0$
Therefore, B is a singular matrix.
As B is singular, its inverse does not exist.