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September 16th, 2014, 08:58 PM   #1
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Invertible Martices

Let $\displaystyle S$ be a set of $\displaystyle d \times d$ matrices of a field of order $\displaystyle q$, such that the difference of any two distinct matrices in $\displaystyle S$ is invertible.

Prove that $\displaystyle |S| \leq q^d.$

I know that the set of $\displaystyle d \times d$ matrices is equal to $\displaystyle q^d$ so this question seems trivial and I don't know where to start.
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September 16th, 2014, 11:12 PM   #2
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The set of dxd matrices would have cardinality $\displaystyle q^{d^2} $ but I don't know where to go from there.
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