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Math Help - [SOLVED] Non-singular Matrix

  1. #1
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    [SOLVED] Non-singular Matrix

    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?
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  2. #2
    tah
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    hi, In such a field, 1=-1 and invertible matrices are exactly the ones with non-zero determinants.
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  3. #3
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    So 1+1=0 does not get used at all?

    And number of non-singular matrices= 6?
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  4. #4
    tah
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    1+1=0 was needed to make sure that K is indeed a field!
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  5. #5
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    true
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