ediculeN

2021-02-25

Determine the null space of each of the following matrices:
$\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]$

Arnold Odonnell

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.
$\left[\begin{array}{cccc}2& 1& |& 0\\ 3& 2& |& 0\end{array}\right]$
${R}_{2}\to {R}_{2}-\frac{3}{2}{R}_{1}$
$=\left[\begin{array}{cccc}2& 1& |& 0\\ 0& \frac{1}{2}& |& 0\end{array}\right]$

We know ${x}_{1},{x}_{2}$ cant be zero since B cant be null
$\therefore 2{x}_{1}+{x}_{2}=0$
Assume ${x}_{2}=C⇒{x}_{1}=\frac{-c}{2}$

Jeffrey Jordon