# Thread: matrix to the power

1. ## matrix to the power

Hi, matrix A=
-7 18
-3 8

I need to find $\displaystyle A^2$ using $\displaystyle A^2=(P)(D^2)(P^-1)$
P=
2 3
1 1

P^-1 =
-1 3
1 -2

D^2 =
4 0
0 1

My problem is how do I multiply out PD^2P^-1 correctly? its doesnt seem to give A^2 as the answer?

Thanks

2. Originally Posted by BadMonkey
Hi, matrix A=
My problem is how do I multiply out PD^2P^-1 correctly? its doesnt seem to give A^2 as the answer?
Let us see. Perhaps the origin of the problem is:

(i) The eigen values of $\displaystyle A$ are $\displaystyle \lambda_1=2,\;\lambda_2=-1$ (both simple), so $\displaystyle A$ is diagonalizable, being:

$\displaystyle D=\begin{bmatrix}{2}&{0}\\{0}&{-1}\end{bmatrix}$

(ii) In fact, the $\displaystyle P$ matrix satisfies:

$\displaystyle P^{-1}AP=D$ or equivalently $\displaystyle A=PDP^{-1}$

(iii) Then,

$\displaystyle A^2=PDP^{-1}PDP^{-1}=PD^2P^{-1}$.

(iv) So,

$\displaystyle A^2=\begin{bmatrix}{2}&{3}\\{1}&{1}\end{bmatrix}\b egin{bmatrix}{4}&{0}\\{0}&{1}\end{bmatrix}\begin{b matrix}{-1}&{3}\\{1}&{-2}\end{bmatrix}=\ldots=\begin{bmatrix}{-5}&{18}\\{-3}&{10}\end{bmatrix}$

Now, find $\displaystyle A^2$ directly. You'll obtain the same matrix.

Regards.