How would I compute $\begin{pmatrix}-5&8\\-4&7\end {pmatrix}^5$ using the relationship between the diagonal matrices and the $n^{\textrm{th}}$ power of a matrix?
My lecturer gave the relation $A^k$=$P^{-1}D^kP$ where $D$ is the diagonalized form of the first matrix. Could somebody give a step by step by step solution to this please?
Best Answer
Imagine that $A$ is diagonalizable, i.e, you can write $A = P^{-1}DP$ for some $P$ and diagonal $D$. Then, $$ A^2 = \left( P^{-1}DP \right) \left( P^{-1}DP \right) = \left( P^{-1}D \left(P P^{-1}\right) DP \right) = \left( P^{-1}D^2 P \right) $$
Can you take this from here?
EDIT In response to the comments below, I am getting $$ P = \pmatrix{1 & -1\\-1 & 2}, P^{-1}= \pmatrix{2 & 1\\1 & 1}, D= \pmatrix{-1 & 0\\0 & 3} $$ so $$ A^5 = \pmatrix{2 & 1\\1 & 1} \pmatrix{-1 & 0\\0 & 243} \pmatrix{1 & -1\\-1 & 2} = \pmatrix{-245 & 488\\-244 & 487} $$