Cardinality of Set S with 2x2 Invertible Matrices from {0,1,2}?

[Oster]

S is the set containing all 2x2 invertible matrices such that the entries come from the the set {0,1,2}. What is the cardinality(number of elements) of this set?

I got 50. Is this correct? What is the best way to go about solving this problem?

 

[daveb]

One way to solve it might be to use the determinant ab-cd where a, b, c, and d, come from the set {0, 1, 2}. Figure out which allows for invertibility, and then how many ways you can get that value for the determinant.

 

[Oster]

Uh I got 42 this way. What's the answer?

 

[SammyS]

What have you tried?

Show some of your work.

Explain some of your thinking.

 

[Oster]

OK! first i tried enumerating the non-degenrate matrices by counting the number of matrices which had one 0 then two 0s then one 1 and no 0s and then two 1s no 0s and last 3 1s and no 0s. I got 50.

Then I looked at all the values the determinant could take (-4 to 4?) and counted the number of matrices which gave the particular value of the determinant. I got 42.

THEN i wrote a program to count the number of degenerate matrices. I got 50. So 50?