Determinants

Date: 05/07/97 at 01:43:24
From: Claudio Meis
Subject: Linear Algebra

If a + b + c + d = 0, how do I show that

    |a b c d|
    |b c d a|  =  0 ?
    |c d a b|
    |d a b c| 

Date: 05/07/97 at 06:00:09
From: Doctor Mitteldorf
Subject: Re: Linear Algebra

Dear Claudio,
    
In a lot of problems, it's easier to think about the properties of the
determinant rather than the forumula or algorithm for computing it.  

One of the fundamental properties is that it is totally ANTISYMMETRIC,
which means that when you interchange any two rows (or two columns), 
the sign of the determinant changes to the opposite of what it was 
before. It follows that if any two rows (or columns) are the same, 
the determinant is zero.  

Another property of the determinant is that it is linear in each row 
or column separately. This means that if you add two vectors to form 
a row, then the determinant of the matrix with the sum of the two 
vectors in place is just the sum of the two determinants with the 
original two vectors in place.

It follows from these two properties that you can add a multiple of 
any row to any other row without changing the determinant  If you 
think about it a little further, you can see that the determinant must 
be zero if the four rows are linearly dependent, that is, if there are 
any numbers m1, m2, m3, and m4 such that: 

m1*(first row) + m2*(second row) + m3*(third row) + m4*(fourth row) = 
0 vector

One last hint: let m1 and m2 and m3 and m4 all be equal to 1/4, and 
apply the theorem about "linearly independent" to your matrix.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/