Question

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

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

Answer 1

Answer: (C)

Here $BC=\begin{bmatrix} 1 & 4 \\ 2 & 9 \end{bmatrix}\begin{bmatrix} 9 & -4 \\ -2 & 1 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}=I$
The given matrix reduce to $A^{3}+A^{5}+A^{7}+\ldots ..+A^{2017}$ which is skew symmetric as $A$ is skew symmetric