$$\left(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5\right)=\left(\begin{array}{rrrrr} 1 & 0 & 3 & 1 & 2 \\ -1 & 3 & 0 & -2 & 1 \\ 2 & 1 & 7 & 2 & 5 \\ 4 & 2 & 14 & 0 & 10 \end{array}\right)$$

$\sim$

$$\left(\begin{array}{rrrrr} 1 & 0 & 3 & 1 & 2 \\ 0 & 3 & 3 & -1 & 3 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 2 & 2 & -4 & 2 \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} 1 & 0 & 3 & 1 & 2 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 3 & 3 & -1 & 3 \\ 0 & 2 & 2 & -4 & 2 \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} 1 & 0 & 3 & 1 & 2 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -4 & 0 \end{array}\right)$$
$\sim$
$$\left(\begin{array}{rrrrr} 1 & 0 & 3 & 1 & 2 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -4 & 0 \end{array}\right)$$

$\sim$
$$\left(\begin{array}{ccccc} \alpha_1 & \alpha_2 & \alpha_3 & \alpha_4 & \alpha_5 \\ 1 & 0 & 3 & 0 & 2 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right),$$

$$\alpha_3=3 \alpha_1+\alpha_2, \alpha_5=2 \alpha_1+\alpha_2 \text {. }$$