number of matrices of rank 3?

Question

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??

score 5 · Answer 1 · answered Sep 27 '14 at 08:18

5

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.

answered Sep 27 '14 at 08:18

Amitai Yuval

19,308

how is the second one -3? – amit Sep 27 '14 at 08:23
There are $3^4$ elements in $\mathbb{F}^4$ and $3$ elements in $span(c_1)$ – Amitai Yuval Sep 27 '14 at 08:25
ohk..ya understood..thanku – amit Sep 27 '14 at 08:37

number of matrices of rank 3?

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