## Matrix A has rank n iff diagnol entries of R' are non zero

Sorry about formatting as this is my first post here.

Question :- A mXn matrix (m>=n), A=Q'R' (Reduced QR)

To show that A has rank n iff diagonal entries of R' are non zero.

My Solution :-
if Ax = 0 => Q'R'x = 0 => R'x =0

let R' = upper triangular matrix (nXn) and x is a column vector.

suppose $r_{nn}$ ..... $r_{k+1k+1}$ are not 0

but $r_{kk} = 0$
since R'x = 0 => $x_n = x_{n-1}..... = x_{k+1} = 0$

but $x_k$ is not zero as $r_{kk}$ is 0 and we can have $x_k$= say 1 ....so we can determine value of $x_{k-1} .... x_1$ by back substitution

=> x is a nonzero vector such that R'x = 0

I am not able to go beyond this.....Am I taking right approach..

Any help is appreciated.