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

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Homework Help Overview

The discussion revolves around determining the cardinality of the set of all 2x2 invertible matrices with entries from the set {0, 1, 2}. The original poster questions the correctness of their initial count of 50 and seeks guidance on the best approach to solve the problem.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the determinant formula to assess invertibility, considering how many matrices yield specific determinant values. There is also an exploration of counting methods based on the presence of zeroes and ones in the matrices.

Discussion Status

Several participants have shared their attempts and results, with one suggesting a determinant-based approach and another providing a count of 42. The discussion reflects a mix of methods and interpretations, with no explicit consensus reached on the correct cardinality.

Contextual Notes

Participants are working under the constraints of the problem, including the specific set of matrix entries and the requirement for invertibility. There is mention of counting degenerate matrices and the range of possible determinant values.

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

Show some of your work.

Explain some of your thinking.
 
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?
 

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