Let $A$ and $B$ be matrices $3\times 3$ with integer entries such that $A,\: A+B,\: A+2B,\: A+3B,\: A+4B,\: A+5B$ and $A+6B $ are all invertible matrices with their inverse have integer entries. Show that $A+2012B$ is invertible and its inverse has integer intries.
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