Answer
(a) The nullspace of $A $ consists of the vectors on the following form
$$x= \left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]= \left[\begin{aligned}0\\0 \end{aligned}\right] .$$
(b) The nullity is $0$.
(c) The rank of $A$ is $0$.
Since $A$ has two columns, one can see that
$$\text{rank}(A)+\text{nullity}(A)=2+0=2.$$
Work Step by Step
Given the matrix
$$
A=\left[ \begin {array}{cc} 1&4\\ 3&2\end {array}
\right].
$$
The reduced row echelon form is
$$
\left[ \begin {array}{cc} 1&0\\ 0&1\end {array}
\right]
.
$$
The corresponding system is
$$
\begin{aligned} x_{1} &=0\\
x_2&=0
\end{aligned}.
$$
The solution of the above system is $x_1= 0$,$x_2=0$.
(a) The nullspace of $A $ consists of the vectors on the following form
$$x= \left[\begin{aligned} x_{1}\\ x_{2} \end{aligned}\right]= \left[\begin{aligned}0\\0 \end{aligned}\right] .$$
(b) The nullity is $0$.
(c) The rank of $A$ is $0$.
Since $A$ has two columns, one can see that
$$\text{rank}(A)+\text{nullity}(A)=2+0=2.$$
