$A=\begin{pmatrix}5 & 2 \\ 4 & 7\end{pmatrix}$ ,find $e^{xA}$

Solution 1:

Pursuant to Edgar's comment, you should diagonalize the matrix, i.e. rewrite $A = PDP^{-1}$ and calculate \begin{align*} e^{At} :&= \sum_{j=0}^{\infty} \frac{(At)^{j}}{j!} \\ &= P \left (\sum_{j=0}^{\infty} \frac{(Dt)^{j}}{j!} \right )P^{-1} \\ &= P\, \mathrm{diag} \{e^{\lambda_{j} t} \} P^{-1}. \end{align*} To do so: first find roots to characteristic equation $\lambda^{2}-12\lambda+27 = 0 \implies \lambda \in \{3,9 \}$ and find eigenvectors $\vec{v_{3}} = (2,-1)^{T}$ and $\vec{v_{9}} = (1,1)^{T}$, whence $P = \begin{pmatrix} 2 & 1 \\ -1 & 1 \end{pmatrix}$.

Finally, one may calculate $P^{-1}$ and $e^{At} = P \begin{pmatrix} e^{3t} & 0 \\ 0 & e^{9t} \end{pmatrix} P^{-1}$ using matrix algebra. I'll leave this as an exercise.