Let A = [ array ll 1 - 1 0 0 & 1 & - 1 0 & 0 & 1 array ] and B = 7...

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

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

Then,

${C}^{2} =$$\begin{array}{ll}0 0 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}$,</p><p>${C}^{3} = \left[\begin{array}{ll}0 0 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}$\right] = {C}^{4} = {C}^{5} = \ldots \ldots .

Now,

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

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

$= 7\left(I + 20C + C2 20 {C}^{2} + . . .\right) - 20\left(I + 7C + C2 7 {C}^{2} + . .\right) + 2I$

$= – 11I + 910{C}^{2}$

= \left[\begin{cases}- 11 & 0 & 910 \\ 0 & - 11 & 0 \\ 0 & 0 & - 11\end{cases}\right]

So,

${b}_{13} = 910$