Give an example of a 3 x 3 matrix A such that Nul(A) # {0}.

Let $T:{\mathrm{ℝ}}^3\to {\mathrm{ℝ}}^3$ be a linear transformation defined by $T(x,y,z)=\left(x-y,y-z,z-x\right)$

Therefore $Ker \left(T\right)=\left\{\left(x,y,z\right)\in R^3: T\left(x,y,z\right)=0\right\}$

$$\begin{gathered} \therefore x-y=0 \\ y-z=0 \\ z-x=0 \\ \therefore x=y=z \end{gathered}$$

$$\begin{gathered} \therefore Ker \left(T\right)=\left\{\left(x,y,z\right)\right\}=\left(1,1,1\right) \\ \therefore Null\left(T\right)=dim\left(Ker\left(T\right)\right) \\ =1\ne 0 \end{gathered}$$

$$\begin{gathered} T(x,y,z)=\left(x-y,y-z,z-x\right) \\ T\left(1,0,0\right)=\left(1,0,-1\right)=a\left(1,0,0\right)+b\left(0,1,0\right)+c\left(0,0,1\right)=\left(a,b,c\right) \\ T\left(0,1,0\right)=\left(-1,1,0\right)=e\left(1,0,0\right)+f\left(0,1,0\right)+g\left(0,0,1\right)=\left(d,e,f\right) \\ T\left(0,0,1\right)=\left(0,-1,1\right)=h\left(1,0,0\right)+i\left(0,1,0\right)+k\left(0,0,1\right)=\left(g,h,i\right) \\ A\left(T\right)=\left[\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right] \\ =\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right] \end{gathered}$$

Then matrix $A$ is 

$A=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ -1& 0& 1\end{array}\right]$