Properties of Determinants

Date: 10/23/2000 at 11:15:05
From: Nooni
Subject: Solving determinants

Dear Dr. Math,

How do we solve determinants and how do we get them in echelon or 
row-echelon form? Are the steps we use based on certain things? Please 
give me an example to help me understand more.

Date: 10/23/2000 at 15:05:26
From: Doctor Rob
Subject: Re: Solving determinants

Thanks for writing to Ask Dr. Math, Nooni.

You cannot "solve" a determinant. A determinant is an expression, and 
you cannot solve an expression, only evaluate it. You can solve 
equations, but a determinant is not an equation. I will assume you 
mean to evaluate a determinant.

Properties of a determinant D:

(a) If the corresponding rows and columns of D are interchanged (that 
is, D is transposed), the value of D is unchanged.

(b) If any two rows (or columns) of D are interchanged, the value of D 
is changed to -D.

(c) If any two rows (or columns) of D are alike, then D = 0.

(d) If each element of a row (or column) of D is multiplied by m, then 
the value of D is changed to m*D.

(e) If to each element of a row (or column) of D is added m times the 
corresponding element in another row (or column), the value of D is 
unchanged.

The process of reducing a determinant to row-echelon form is a series 
of applications of these rules, especially (b), (d), and (e), using 
rows, to create a determinant whose value is a known multiple of the 
value of D, and which has zeroes below the main diagonal. Then the 
value of this new determinant is the product of the entries on the 
main diagonal. From that, you can figure out the value of D.

Example. Evaluate the following determinant:

         | 2  1 -2|
     D = | 1  1  1|
         |-1 -2  3|

Add -1/2 times the first row to the second row (Property (e)):

         | 2   1  -2|
     D = | 0  1/2  2|
         |-1  -2   3|

Add 1/2 times the first row to the third row ((e) again):

         |2   1  -2|
     D = |0  1/2  2|
         |0 -3/2  2|

Add 3 times the second row to the third row ((e) again):

         |2   1  -2|
     D = |0  1/2  2|
         |0   0   8|

Now D is in row-echelon form. When it is in this form, the value of D 
is the product of the entries on the main diagonal, so

     D = 2*(1/2)*8 = 8

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