Exercise 12

Prove that $$\alpha e^{\lambda t} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \beta e^{\lambda t} \begin{pmatrix} t \\ 1 \end{pmatrix}$$ is the general solution of $${\mathbf x}' = \begin{pmatrix} \lambda & 1 \\ 0 & \lambda \end{pmatrix} {\mathbf x}.$$