Linear algebra riddle?

[Mozly]

This is a bonus on my practice exam. Can you help me solve it?


Amy and Brian play a game in which they take turns filling the entries of an initially
empty 2010×2010 matrix. Amy plays first. At each turn, a player chooses a real number
and places it in a vacant entry. The game ends when the matrix is completely filled. Amy
wins if the resulting matrix is invertible, and Brian wins if the resulting matrix is not
invertible.
Which player has a winning strategy? Explain.

 

[Todd]

Brian. To make the matrix uninvertible, he needs to ensure that one of the columns exactly equals another column. So consider columns 1 and 2. If Amy puts a number in one of these columns, Brian will put that same number next to it in the other column. If Amy puts a number in any other column, Brian puts any number in any column other than 1 and 2. Because there are an even number of entries in columns 3 through 2010, Brian will never be forced to play in columns 1 or 2 if Amy doesn't do it first.