# Thread: Matrix to the Power 10

1. ## Matrix to the Power 10

I have to calulate A^10.

I have been given A =
(1 2
2 1)

and B =
(1 1
1 -1)

I have calculated that B^-1AB =
(3 0
0 1)

and I don't really understand what my notes mean to take it on from here. Help would be appreciated.

2. To find $\displaystyle A^{10}$

You must find a diagonal matrix $\displaystyle D$ such that

$\displaystyle A^n= BD^nB^{-1} \implies D^n= B^{-1}A^nB$

In your case $\displaystyle n=10$

and where $\displaystyle B$ is the eigenvectors of $\displaystyle A$

So first thing to do is find the eigenvalues and eigenvectors of $\displaystyle A$

It looks like you have been given that so use

$\displaystyle A^n= BD^nB^{-1}$

As you have $\displaystyle D , B$ and $\displaystyle B^{-1}$

3. If that was a little to hard to follow then,

$\displaystyle \left( {\begin{array}{*{20}{c}} 1& 2 \\ 2 & 1\\ \end{array}} \right) ^{10} = \left( {\begin{array}{*{20}{c}} 1 & 1 \\ {-1} & 1 \\ \end{array}} \right) \left( {\begin{array}{*{20}{c}} 3 & 0 \\ 0 & 1 \\ \end{array}} \right) ^{10} \left( {\begin{array}{*{20}{c}} 1 & 1 \\ {-1} & 1 \\ \end{array}} \right) ^{-1}$

Just solve the RHS.