[SOLVED] Non-singular Matrix

**nmatthies1** · February 26th 2009, 01:41 AM

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 $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ (det=0), $\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ (det=-1), $\begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}$ (det=0), $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ (det=1), $\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}$ (det=0), $\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}$ (det=0), $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$ (det=-1), $\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$ (det=-1), $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$ (det=1), $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ (det=1), $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ (det=1-1).

now which ones are non-singular, considering that 1+1=0?

**tah** · February 26th 2009, 02:25 AM

hi, In such a field, 1=-1 and invertible matrices are exactly the ones with non-zero determinants.

**nmatthies1** · February 26th 2009, 02:32 AM

So 1+1=0 does not get used at all?

And number of non-singular matrices= 6?

**tah** · February 26th 2009, 02:42 AM

1+1=0 was needed to make sure that K is indeed a field!

**nmatthies1** · February 26th 2009, 02:47 AM

true

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