Matrix e^(tA)

[borkborkmath]

I'm trying to figure out if I am doing this correctly.

Given the matrix A:
[1 -.5]
[.5 2]
write down e^(tA) and calculate e^(tA)X, X = (1,1)

So far what I did was, I found the eigenvalues (1.5 repeated) and the eigenvectors ( [-1 1], [1 1]). (The eigenvectors are transposed)
Then I set e^(tA) = T^(-1)AT, where T = [eigenvector1, eigenvector 2]

Which leads to e^(tA):
[1.75 .75]
[0 1.5]


I Feel that I do not have the correct equation for e^(tA), can anyone help me?

 

[galactus]

\(\displaystyle e^{At}=1+\begin{bmatrix}1&-1/2\\1/2&2\end{bmatrix}t+\begin{bmatrix}1&-1/2\\1/2&2\end{bmatrix}\frac{t^{2}}{2!}+........+\begin{bmatrix}1&-1/2\\1/2&2\end{bmatrix}\frac{t^{n}}{n!}+............=\sum_{n=0}^{\infty}\begin{bmatrix}1&-1/2\\1/2&2\end{bmatrix}\frac{t^{n}}{n!}\)