If $\mathrm{A}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$, then

$\begin{array}{l}{\mathrm{A}}^2=\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]\\ {\mathrm{A}}^3=\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]=\left[\begin{array}{lll}1& 0& 0\\ 2& 0& 1\\ 1& 1& 0\end{array}\right]\\ {\mathrm{A}}^3-{\mathrm{A}}^2=\left[\begin{array}{ccc}0& 0& 0\\ 1& -1& 1\\ 0& 1& -1\end{array}\right]\end{array}$

and $\mathrm{A}-\mathrm{I}=\left[\begin{array}{ccc}0& 0& 0\\ 1& -1& 1\\ 0& 1& -1\end{array}\right]$

$\Rightarrow {\mathrm{A}}^3-{\mathrm{A}}^2=\mathrm{A}-\mathrm{I}$                                (1)

$\begin{array}{c}\mathrm{Now}, \det \left({\mathrm{A}}^{\mathrm{n}}-\mathrm{I}\right)=\det \left((\mathrm{A}-\mathrm{I})\left(\mathrm{I}+\mathrm{A}+{\mathrm{A}}^2+\dots +{\mathrm{A}}^{\mathrm{n}-1}\right)\right)\\ =\det .(\mathrm{A}-\mathrm{I})\times \det .\left(\mathrm{I}+\mathrm{A}+{\mathrm{A}}^2+\dots +{\mathrm{A}}^{\mathrm{n}-1}\right)\\ =0\end{array}$

From (1), A⁴ - A³ = A² - A                         (2)

Again from (1), A⁵ - A⁴ = A³ - A² = A - I

Thus, if n is even, Aⁿ - A^n-1 = A² - A          (3)

If n is odd, Aⁿ - A^n-1 = A - I                        (4)

Consider that n is even.

$\therefore {\mathrm{A}}^{\mathrm{n}}-{\mathrm{A}}^{\mathrm{n}-1}={\mathrm{A}}^2-\mathrm{A}$            [from(3)]

and ${\mathrm{A}}^{\mathrm{n}-1}-{\mathrm{A}}^{\mathrm{n}-2}=\mathrm{A}-\mathrm{I}$            [from(4)]

Adding these, we get

${\mathrm{A}}^{\mathrm{n}}-{\mathrm{A}}^{\mathrm{n}-2}={\mathrm{A}}^2-\mathrm{I}$

$\begin{array}{l}\Rightarrow {\mathrm{A}}^{\mathrm{n}}={\mathrm{A}}^{\mathrm{n}-2}+{\mathrm{A}}^2-\mathrm{I}\\ =\left({\mathrm{A}}^{\mathrm{n}-4}+{\mathrm{A}}^2-\mathrm{I}\right)+{\mathrm{A}}^2-\mathrm{I}\\ =\left({\mathrm{A}}^{\mathrm{n}-6}+{\mathrm{A}}^2-\mathrm{I}\right)+2\left({\mathrm{A}}^2-\mathrm{I}\right)\\ \cdots \dots \dots \\ \cdots \cdots \dots \\ =\left({\mathrm{A}}^2\right)+\frac{\mathrm{n}-2}{2}\left({\mathrm{A}}^2-\mathrm{I}\right)\end{array}$

$\begin{array}{l}\therefore {\mathrm{A}}^{\mathrm{n}}=\left(\frac{\mathrm{n}}{2}\right){\mathrm{A}}^2-\left(\frac{\mathrm{n}-2}{2}\right)\mathrm{I}\\ \therefore {\mathrm{A}}^{200}=100{\mathrm{A}}^2-99\mathrm{I}\end{array}$

                 $=100\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]-99\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 100& 1& 0\\ 100& 0& 1\end{array}\right]$