# 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 $A^{10}$

You must find a diagonal matrix $D$ such that

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

In your case $n=10$

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

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

It looks like you have been given that so use

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

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

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

$
\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.