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### Properties of Determinants

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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/
```
Associated Topics:
College Linear Algebra
High School Linear Algebra

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