DeterminantsDate: 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/ |
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