Let $\mathrm P$ be a $2 \times 2$ matrix such that $\mathrm P^{102}= \mathrm O$

Question

Can someone guide me how to deal with these problems. Is there an organised way to proceed in solving these type of questions . Thank you.

Answer 1

Show that $0$ is the only eigenvalue. Since zero is the only eigenvalue, you can show it is similar to the following matrix:

$$\begin{pmatrix} 0 & x \\ 0 & 0 \end{pmatrix}$$

for some $x$.

From here you can look at each of those cases.