Statement
"The rank of a matrix does not change by pre-multiplication
or post multiplication with a non-singular matrix"
Doubt
Now, Let P and Q are two non-singular matrices of rank M and N. Then as per the statement:
on post multiplying Q with P : $ \rho(P*Q)= M $
on pre-multiplying Q with P: $ ρ(Q*P)= M$ -------------(1)
Similarly, on post and pre multiplying P with Q: $ρ(Q*P)= N$ and $ρ(Q*P)= N$ -----------(2)
from 1 and 2, M=N , which means every Non-singular matrix will have same rank.
I know it's not correct, but where I am wrong ?