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

Skilled2021-02-26Added 109 answers

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.

$\left[\begin{array}{cccc}2& 1& |& 0\\ 3& 2& |& 0\end{array}\right]$

${R}_{2}\to {R}_{2}-\frac{3}{2}{R}_{1}$

$=\left[\begin{array}{cccc}2& 1& |& 0\\ 0& \frac{1}{2}& |& 0\end{array}\right]$

$2{x}_{1}+{x}_{2}=0\text{or}\frac{1}{2}{x}_{2}=0$

We know${x}_{1},{x}_{2}$ cant be zero since B cant be null

$\therefore 2{x}_{1}+{x}_{2}=0$

Assume${x}_{2}=C\Rightarrow {x}_{1}=\frac{-c}{2}$

$\therefore \text{Null space}B=\left[\begin{array}{c}-\frac{c}{2}\\ c\end{array}\right]\mathrm{\forall}c\in \mathbb{R}$

The null space of matrix A is B such that AB=0 for all B is not null.

Given

Step 2

To findout nullspace , writing the matrix in Augment form and convert the, to echelon form.

We know

Assume

Jeffrey Jordon

Expert2022-01-27Added 2575 answers

Answer is given below (on video)