<s>[INST] Find $A^{100}+A^5$ Let $A= \begin{pmatrix} 1&-1 \\ 2&-2 \\  \end{pmatrix}$. We are to find $A^{100} +A^5$
My attempt : I found characteristic equation , $x^2+x=0$, so eigenvalues are $0$ and $-1$ hence its diagnol matrix is $$D=  \begin{pmatrix} 0&0 \\ 0&-1 \\  \end{pmatrix}$$  So $D^{100}+D^5$ is null matrix.
But I am getting something else if I use method given here To Find $A^{50}$ by achille hui.
Please help in this matter. Thanks
 [/INST] Observe the following computation:
$$
A^2 \;\; =\;\; \left [ \begin{array}{cc}
1 & -1 \\
2 & -2\\
\end{array} \right ]\left [ \begin{array}{cc}
1 & -1 \\
2 & -2\\
\end{array} \right ] \;\; =\;\; \left [ \begin{array}{cc}
-1 & 1 \\
-2 & 2\\
\end{array} \right ].
$$
Similarly if we compute $A^3$ we find
$$
A^3 \;\; =\;\; \left [ \begin{array}{cc}
-1 & 1 \\
-2 & 2\\
\end{array} \right ]  \left [ \begin{array}{cc}
1 & -1 \\
2 & -2\\
\end{array} \right ] \;\; =\;\;  \left [ \begin{array}{cc}
1 & -1 \\
2 & -2\\
\end{array} \right ]. 
$$
Therefore we have that $A^{2n} = -A$ and $A^{2n+1} = A$ for all $n \in \mathbb{N}$.  We conclude that 
$$
A^{100} + A^5 \;\; =\;\; -A + A \;\; =\;\; \textbf{0}_{2\times 2}.
$$
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