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How does this change the determinant of the Matrix?

If I have the matrix (M) with the determinant det(M), and then i have the matrix M' which is:

[            |0,0]

[   M      |0,0]

[            |0,0]

[0, 0, t-1|,-1,0]

[0, 0, t-1|,-1,t]

would you have to take into consideration the [t-1]s? since they are directly "under" the M matrix. I put this into my calculator using placeholder numbers, and I got the determinant of M to be 9, and M' to be -9, when i substituted a -1 diagonally across from the original M matrix, even though I also added terms below the matrix. So although I'm confident that it doesn't change the determinant, I'm wondering WHY it doesn't.

2 Answers

  • 2 months ago
    Favorite Answer

    Find the determinant of M' by using cofactor expansion on the last column of M'

    You will get  t times the determinant of the matrix below.

    [            |0]

    [   M      |0]

    [            |0]

    [0, 0, t-1|,-1]

    Now use cofactor expansion on the last column again.

    The determinant of M' will be -t*det(M)

    • Spencer2 months agoReport

      hi, could you detail the steps of the cofactor expansion? basically, i'm punching way above my weight here considering I've learned nothing about matrices in lessons: i'm trying to learn how to do determinants by myself, so a list of steps of the cofactor process would be appreciated, thanks!

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  • Anonymous
    2 months ago

    Question is not clear,  But assuming det(M) is not zero, then changing the value of t *does* change det(M').

    (If det(M) = 0 then det(M') = 0 also.)

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