Let A be an nxn marix, if A is row equivalent to a matrix B and there is a non-zero matrix C such that BC = 0n, prove that A is singular.
A row equivalent to B $\displaystyle \Longrightarrow\,\exists $ elementary matrices $\displaystyle E_1,...,E_k\,\,s.t.\,\,E_1...E_kA=B$ , so:
$\displaystyle 0_n=BC=(E_1....E_kA)C$ , and if A weren't singular and since elementary matrices aren't singular, multiplying the right hand from the left by $\displaystyle A^{-1}E_K^{-1}...E_1^{-1}$ you get a contradiction.
Tonio
if A weren't singular and since elementary matrices aren't singular, multiplying the right hand from the left by $\displaystyle A^{-1}E_K^{-1}...E_1^{-1}$ you get a contradiction.
what does does this mean? are we not trying to prove that A is singular? there shouldnt be a contradction?