Let $A=\left(\begin{array}{ccc}1 & -1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1\end{array} ight)$ and $B=7 A^{

Question

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

Professor PC Thomas Classes · Answer

(910) : Let $A=\left(\begin{array}{ccc}1 & -1 & 0 \ 0 & 1 & -1 \ 0 & 0 & 1\end{array}ight)=I+P$, where,\ $I=\left(\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}ight), P=\left(\begin{array}{ccc}0 & -1 & 0 \ 0 & 0 & -1 \ 0 & 0 & 0\end{array}ight)$\ $P^{2}=\left[\begin{array}{lll}0 & 0 & 1 \ 0 & 0 & 0 \ 0 & 0 & 0\end{array}ight], P^{3}=\left[\begin{array}{lll}0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0\end{array}ight]$\ Now, $B=7 A^{20}-20 A^{7}+2 I=7(I+P)^{20}$\ $-20(I+P)^{7}+2 I$\ $=7\left(I+20 P+{ }^{20} C_{2} P^{2}ight)-20\left(I+7 P+{ }^{7} C_{2} P^{2}ight)+2 I$\ $	herefore \quad b_{13}=7 	imes{ }^{20} C_{2}-20 	imes{ }^{7} C_{2}=910$\