Mathematics - Matrix Question with Solution | TestHub

Given the equation $P^2=I-P$ (1).

We can derive further powers of $P$:

$$\begin{array}{rl}{P}^3& =P(I-P)\\ P^3& =P-P^2\\ P^3& =P-(I-P)\\ P^3& =2P-I\end{array}$$

Similarly, we can find $P^4$, $P^5$, and $P^6$:

$$\begin{array}{rl}{P}^4& =2P^2-P\\ P^5& =2P^3-P^2=2(2P-I)-(I-P)=4P-2I-I+P=5P-3I\\ P^6& =5P^2-3P=5(I-P)-3P=5I-5P-3P=5I-8P\end{array}$$

Therefore, for $P^6=5I-8P$, we have $n=6$.