Calculating Matrix Determinants by Expansion of Minors

Date: 11/19/2003 at 21:30:57
From: Russ
Subject: Determinants of 3x3 matrices

The matrix I am working on is

   1  4  6
  -7  8  4
  -1  2 -3

When I find the determinant of this 3x3 by diagonals I get -168.  When 
I use expansion of minors by row 1 I get -168, and in row 3 I also get
-168.  But when I use row 2 I get 168.  Every problem I check I always
get row 2 to be the opposite.  I just do not understand why.  Could
someone please explain this?

Date: 11/19/2003 at 23:47:19
From: Doctor Jordan
Subject: Re: Determinants of 3x3 matrices

Dear Russ,

You can think of a matrix like a checkerboard when you're calculating
with cofactors.  Consider,

  ---------
  | + - + |  
  | - + - |
  | + - + |
  --------|

If you're taking a cofactor from an entry in the matrix, you can look
at the checkerboard representation to find if this cofactor has a
coefficient of 1 or -1.  For example, if you're taking cofactors from
the first row, you would would have the determinant equal to

  (1)*cofactor from column 1 multiplied by its minor determinant +
  (-1)*cofactor from column 2 multiplied by its minor determinant +
  (1)*cofactor from column 3 multiplied by its minor determinant.

But if, like in your problem, you're taking the cofactors from the
second row, the determinant is equal to

  (-1)*cofactor from column 1 multiplied by its minor determinant +
  (1)*cofactor from column 2 multiplied by its minor determinant +
  (-1)*cofactor from column 3 multiplied by its minor determinant.

Do you see the difference?  We're taking the coefficients of the
cofactors from the entries in the checkerboard, and for row two _the
coefficients (entries) are different than from row 1_.  Can you see
why not considering this aspect, as you are doing, would give you the
exact opposite result?

Does that make sense?  If you have any questions, write back.  Good luck!

- Doctor Jordan, The Math Forum
  http://mathforum.org/dr.math/