Science:Math Exam Resources/Courses/MATH152/April 2015/Question A 27/Hint 1

Compute the eigenvalues and eigenvectors of the coefficient matrix

$A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.$

If $A$ has two eigenvalues $\lambda _{1},\lambda _{2}$ with associated eigenvectors $\mathbf {v} _{1},\mathbf {v} _{2}$, then

${\begin{bmatrix}x\\y\end{bmatrix}}=c_{1}e^{\lambda _{1}t}\mathbf {v} _{1}+c_{2}e^{\lambda _{2}t}\mathbf {v} _{2}$

for some constants $c_{1},c_{2}$.