Question

Determine the null space of each of the following matrices:
\(\begin{bmatrix}2 & 1 \\3 & 2 \end{bmatrix}\)

Answer 1

Step 1
The null space of matrix A is B such that AB=0 for all B is not null.
Given \(A=\begin{bmatrix}2 & 1 \\3 & 2 \end{bmatrix}\)
\(B=\begin{bmatrix}x_1 \\ x_2 \end{bmatrix}\)
\(\begin{bmatrix}2 & 1 \\3 & 2 \end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \end{bmatrix}=\begin{bmatrix}0 \\ 0 \end{bmatrix}\)
Step 2
To findout nullspace , writing the matrix in Augment form and convert the, to echelon form.
\(\begin{bmatrix}2 & 1&|&0 \\3 & 2&|&0 \end{bmatrix}\)
\(R_2 \rightarrow R_2 - \frac{3}{2}R_1\)
\(=\begin{bmatrix}2 & 1&|&0 \\0 & \frac{1}{2}&|&0 \end{bmatrix}\)
\(2x_1+x_2=0 \text{ or } \frac{1}{2}x_2=0\)
We know \(x_1,x_2\) cant be zero since B cant be null
\(\therefore 2x_1+x_2=0\)
Assume \(x_2=C \Rightarrow x_1=\frac{-c}{2}\)
\(\therefore \text{ Null space } B=\begin{bmatrix}-\frac{c}{2} \\ c \end{bmatrix} \forall c \in \mathbb{R}\)