Question

What is the number of non-singular matrices over , the finite field with two elements?

Answer 1

Answer:

I have to find out the number of 3×3 non singular matrices over a field of order 2. I tried in the following way. First to find out a non singular matrix A, clearly any row of A can't be full of 0s. So the first row (say) can be filled up by (8−1) ways. Once the row is filled up,the next row can't be the same and also can't be full of zeros,so we can fill the next row by (8−2) ways. And at last the third row also can't be full of zeros,same as the first row,and same as the second row also.So we have (8−3) choices. Hence the number of non singular matrices seems to be 7×6×5=210. Am I right? Or there are more non singular matrices ? May be less also. Please correct me if I am wrong. Thank you.

Explanation:

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