number of matrices of rank 3? Let M be the space of all $4\times 3$ matrices with entries in the finite field of three elements.The number of matrices of rank 3 in M are?
A. $(3^4-3)(3^4-3^2)(3^4-3^3)$
B.$(3^4-1)(3^4-2)(3^4-3)$
C.$(3^4-1)(3^4-3)(3^4-3^2)$
D.$(3^4)(3^4-1)(3^4-2)$
which of the following is the answer??

The first column can be any non-zero vector in $\mathbb{F}^4$. There are $3^4-1$ of those. Having chosen one of them, call it $c_1$, and then the second column needs to be any vector not in $span(c_1)$. There are $3^4-3$ of those. The third column needs to be outside $span(c_1,c_2)$, so it is one of $3^4-3^2$ options.
The answer is thus C.
