Counting Invertible Matrices in GL(3, Z_2)

[Treadstone 71]

Is there a systematic way of counting the number of invertible matrices in a general linear group with entries in a finite ring? For example, GL(3, Z_2). The determinant has to be zero, but other than that, I don't know any systematic way of counting them. I usually start by saying that there are at least 13 non-invertible ones (if at least one row or column are zeros) then I look at the equation of the determinant and try to go from there.

 

[matt grime]

As long as you mean 'over a finite field' the answer is yes, since invertible is the same as linear independence of rows (or columns)

 

[tacman]

Yes, there is a systematic way of counting the number of invertible matrices in a general linear group with entries in a finite ring. This is known as the order of the group, and it is a fundamental concept in group theory.

In the case of GL(3, Z_2), the order of the group is given by the formula |GL(3, Z_2)| = (2^3 - 1)(2^3 - 2)(2^3 - 2^2) = 168. This means that there are 168 matrices in GL(3, Z_2) that are invertible.

To understand how this formula is derived, we need to first understand the concept of a determinant. In general, the determinant of a square matrix is a scalar value that represents the scaling factor of the matrix. For a matrix to be invertible, its determinant must be non-zero.

In the case of GL(3, Z_2), the determinant must be non-zero, but it is also limited to the values 0 and 1 since Z_2 is a finite ring. This means that we can only have 2^3 - 1 = 7 possible values for the determinant. Additionally, the determinant must be non-zero, so we can eliminate the value 0, leaving us with 6 possible values for the determinant.

Next, we need to consider the possible values for the entries of the matrix. Since we are working with Z_2, the entries can only be 0 or 1. This means that for each of the 3 rows in the matrix, there are 2^3 = 8 possible combinations of entries. However, we must also account for the fact that the determinant is non-zero, so we must eliminate the rows that would result in a determinant of 0. This leaves us with 2^3 - 1 = 7 possible combinations of entries for each row.

To find the total number of invertible matrices, we multiply the number of possible determinant values (6) by the number of possible combinations of entries for each row (7^3). This gives us the formula |GL(3, Z_2)| = (2^3 - 1)(2^3 - 2)(2^3 - 2^2) = 168.

In summary, the systematic way of counting the number of invertible matrices in GL(3, Z_2)