# Solve the Following Questions

Question:

Let $\mathrm{A}=\left(\begin{array}{rrr}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$ and $\mathrm{B}=7 \mathrm{~A}^{20}-20 \mathrm{~A}^{7}+2 \mathrm{I}$,

where $I$ is an identity matrix of order $3 \times 3$. If $B=\left[b_{i j}\right]$, then $b_{13}$ is equal to

Solution:

Let $\mathrm{A}=\left(\begin{array}{ccc}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)=\mathrm{I}+\mathrm{C}$

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

$C^{2}=\left(\begin{array}{lll}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$

$C^{3}=\left(\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)=C^{4}=C^{5}=\ldots \ldots$

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

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

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

So

$\mathrm{b}_{13}=7 \times{ }^{20} \mathrm{C}_{2}-20 \times{ }^{7} \mathrm{C}_{2}=910$