We introduce a new technique for proving formula size lower bounds based on
matrix rank. A simple form of this technique already gives bounds at least as
large as those given by the method of Khrapchenko, originally used
to prove an $n^2$ lower bound on the parity function. We also apply our method to the parity function, and give an exact expression for the formula size of
parity: if $n=2^\ell +k$, where $0\le k < 2^\ell$, then the formula size of
parity on $n$ bits is exactly $2^\ell(2^\ell+3k)=n^2+k2^\ell-k^2$. Such a
bound cannot be proven by any of the lower bound techniques of Khrapchenko,
Ne\v{c}iporuk, Koutsoupias, or the quantum adversary
method, which are limited by $n^2$.