row reduced form with element $i$

Question

find rank(A)

find basis of$ R(L_A)$ consisting of column vectors

find basis of $N(L_A)$ $$A=\left[\begin{array}{ccc}0&i&-1\\1+i&1&1+2i\\1-i&2&1+i\\-i&1-i&1\end{array}\right]$$ i figure out the reduce form is $$A=\left[\begin{array}{ccc}1&0&1\\0&1&i\\0&0&0\\0&0&0\end{array}\right]$$, so the rank(A)=2, but what is basis of $R(L_A)$ and$N(L_A)$

Answer 1

basic of A is $$N(A)=\left[\begin{array}{c}1\\i\\1\\\end{array}\right]$$ and basis of R(A) is $$R(A)=\left[\begin{array}{cc}0&i\\1+i&1\\1-i&2\\-i&1-i\end{array}\right]$$, am i doing right