## anonymous 5 years ago please help guys A= [0 0 -2;1 2 1;1 0 3] find a matrix P diagonazlizes A. what is A^30

If $$D$$ is the diagonalization of $$A$$, then we can write:$A=PDP^{-1}$where:$P=\left[\begin{matrix}e_1 & e_2 & e_3\end{matrix}\right]$and $$e_i$$ represents the i'th eigenvector of $$A$$. You first need to find the eigenvalues of A using:$\left|A-\lambda I\right|=0$solving this should give you the three eigenvalues of $$A$$, $$\lambda_1, \lambda_2, \lambda_3$$. Then use each of the eigenvalues to solve:$(A-\lambda I)x=0$this should give you the three eigenvectors $$e_1, e_2, e_3$$. Hence you have found $$P$$. Ti find $$A^{30}$$ you can use the property:$A^n=(PDP^{-1})^n=PDP^{-1}*PDP^{-1}*PDP^{-1}*...$and note that on either side of the internal multiplications you have $$P^{-1}*P=I$$ and therefore these collapse to give:$A^n=PD^nP^{-1}$and calculating $$D^n$$ for the diagonal matrix $$D$$ is trivial.