Determine the null space of each of the following matrices: [2132]

Step 1
The null space of matrix A is B such that AB=0 for all B is not null.
Given $A=\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]$
$B=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]$
$\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$
Step 2
To findout nullspace , writing the matrix in Augment form and convert the, to echelon form.
$\left[\begin{array}{cccc}2& 1& |& 0\\ 3& 2& |& 0\end{array}\right]$
$R_2\to R_2-\frac{3}{2}{R}_1$
$=\left[\begin{array}{cccc}2& 1& |& 0\\ 0& \frac{1}{2}& |& 0\end{array}\right]$
$2x_1+x_2=0\text{ or }\frac{1}{2}{x}_2=0$
We know $x_1,x_2$ cant be zero since B cant be null
$\therefore 2x_1+x_2=0$
Assume $x_2=C\Rightarrow x_1=\frac{-c}{2}$
$\therefore \text{ Null space }B=\left[\begin{array}{c}-\frac{c}{2}\\ c\end{array}\right]\mathrm{\forall }c\in \mathbb{R}$

Answer is given below (on video)