Question

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

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

2021-02-26
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 $$x_2=C \Rightarrow x_1=\frac{-c}{2}$$
$$\therefore \text{ Null space } B=\begin{bmatrix}-\frac{c}{2} \\ c \end{bmatrix} \forall c \in \mathbb{R}$$