Question

If $A$ is a skew symmetric matrix of order $2$ and $B,C$ are matrices $\left[\begin{array}{cc}1& 4\\ 2& 9\end{array}\right],\left[\begin{array}{cc}9& -4\\ -2& 1\end{array}\right],$ then $A^{3}BC+A^5\left(B^2{C}^2\right)+A^7\left(B^3{C}^3\right)+\dots ..A^{2017}\left(B^{1009}{C}^{1009}\right)$ is

• A. a symmetric matrix
• B. an identity matrix
• C. a skew symmetric matrix
• D. None of these

Answer 1

Answer: (C)

Here $BC=\left[\begin{array}{cc}1& 4\\ 2& 9\end{array}\right]\left[\begin{array}{cc}9& -4\\ -2& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$
The given matrix reduce to $A^3+A^5+A^7+\dots ..+A^{2017}$ which is skew symmetric as $A$ is skew symmetric