If A is skew-symmetric matrix of order $2 and B, C$ are matrices $\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}9\end{array}\right]\text{ and }\left[\begin{array}{cc}9& -4\\ -2& 1\end{array}\right]$ respectively,If X=$A^{3}BC+A^5\left(B^2{C}^2\right)+A^7\left(B^3{C}^3\right)+\dots \dots +A^{2n+1}{B}^n{C}^n$ then X is

 We observe that BC =I =CB, 

$$\begin{gathered} BC=\left[\begin{array}{l}1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}4\\ 2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}9\end{array}\right]\left[\begin{array}{cc}9& -4\\ -2& 1\end{array}\right] \\ =\left[\begin{array}{cc}9-8& -4+4\\ 18-18& -8+9\end{array}\right] \\ =\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$

$\text{ ∴ }{\mathrm{B}}^n{\mathrm{C}}^n={\mathrm{C}}^n{B}^n=I$ for all $n\in N$

Let $X=A^{3}BC+A^5\left(B^2{C}^2\right)+A^7\left(B^3{C}^3\right)+\dots +A^{2n+1}{B}^n{C}^n.$

Then, $X=A^{3}I+A^{5}I+A^{7}I+\dots +A^{2n+1}I$

$\Rightarrow X=A^3+A^5+A^7+\dots +A^{2n+1}$

$\Rightarrow X^T={\left(A^3+A^5+A^7+\dots +A^{2n+1}\right)}^T$

$\Rightarrow X^T={\left(A^3\right)}^T+{\left(A^5\right)}^T+{\left(A^7\right)}^T+\dots +{\left(A^{2n+1}\right)}^T$

$\Rightarrow X^T={\left(A^T\right)}^3+{\left(A^T\right)}^5+{\left(A^T\right)}^7+\dots +{\left(A^T\right)}^{2n+1}$

$\Rightarrow X^T=(-A{)}^3+(-A{)}^5+(-A{)}^7+\dots +(-A{)}^{2n+1}$

                                                            $\left[\because A^T=-A\right]$

$\Rightarrow X^T=-X\Rightarrow X$   is a skew-symmetric matrix.