Let $A=\left[\begin{array}{c}\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\end{array}\right]$ and $B=7A^{20}-20A^7+2I$,
where $I$ is an identity matrix of order $3\times 3.$ If $B=\left[b_{ij}\right]$, then $b_{13}$ is equal to

Let $A=\left[\begin{array}{c}\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\end{array}\right]=I+C$

where $I=\left[\begin{array}{c}\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\end{array}\right], C=\left[\begin{array}{c}\begin{array}{ccc}0& -1& 0\\ 0& 0& -1\\ 0& 0& 0\end{array}\end{array}\right]$

Then,

$C^2=\begin{array}{c}\left[\begin{array}{c}\begin{array}{lll}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\end{array}\right]\end{array}$,

$C^3=\left[\begin{array}{c}\begin{array}{lll}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\end{array}\right]=C^4=C^5=\dots \dots .$

Now,

$B=7{ A}^{20}-20{ A}^7+2I$

$=7(I+C{)}^{20}-20(I+C{)}^7+2I$

$=7\left(I+20C+{}^{20}C_2{C}^2+...\right)-20\left(I+7C+{}^{7}C_2{C}^2+..\right)+2I$

$=–11I+910 C^2$

$=\left[\begin{array}{ccc}-11& 0& 910\\ 0& -11& 0\\ 0& 0& -11\end{array}\right]$

So,

$b_{13}=910$