The determinant of any $3\times 3$ matrix of rank $2$ is $0$, and generalization

Question

Here i want to show The determinant of of any $3\times 3$ matrix of rank $2$ is $0$. Can anyone give me a hint or proof for this?

Further it is generalized to: for any $n\times n$ matrix of rank $k$, all the $(k+1)$-minors vanish.

Please give me some hint or prove of above statements. Thanks

Answer 1

The rank two condition on $A$ means that the Gauss eliminations gives $$ A\sim \left(\matrix{* & * & *\\ 0 & * & *\\ 0 & 0 & 0}\right). $$ The Gauss eliminations does not change the determinant, therefore $$ \det A=\det\left(\matrix{* & * & *\\ 0 & * & *\\ 0 & 0 & 0}\right)=0. $$