hi, In such a field, 1=-1 and invertible matrices are exactly the ones with non-zero determinants.
Let K be a finite field with only the two elements 0 and 1, where 1+1=0.
(i) How many 2 x 2 matrices with entries in K are there?
Ok so for this I expect the answer to be the number of different ways one can write a 4 digit combination of the numbers 1 and 0 and so my answer is 16. But it just seems so trivial...does it seem right?
(ii) How many of these are non-singular?
Now for this I know that a matrix is non-singular if it is invertible, which is the case when the determinant of the matrix is not equal to 0. But the thing is, I got quite confused about the part where 1+1=0. How do I factor this in?
Now all the possible matrices are (det=0), (det=0), (det=0), (det=0), (det=0), (det=0), (det=-1), (det=0), (det=1), (det=0), (det=0), (det=-1), (det=-1), (det=1), (det=1), (det=1-1).
now which ones are non-singular, considering that 1+1=0?