Calculating Determinants 101

You' have to learn

the thr rules of

determinants.

Thr

rules?

Yep, the terms in

the determinant

formula are formed

aording to

certainrules.

Take a closer lk at

the term indexes.

Pay special

aention to the

left index in each

factor.

The left

side...

Flllip

?

Rule

1

102 Chapter 4 More Matrices

Oh, they a go

from one to

the number of

dimensions!

Exa ctly.

And that's rule

number one!

Now for the right

indexes.

H...

They sm

a bit more

random.

Actuay, they're not. Their orders are a

permutations of 1, 2, and 3—like in the table

to the right. This is rule number two.

I s it

now!

Permutations of 1–2

Permutations of 1–3

Paern 1

Paern 2

Paern 1

Paern 2

Paern 3

Paern 4

Paern 5

Paern 6

Rule

2

Calculating Determinants 103

The third rule

is a bit tricky,

so don't lose

concentration.

Okay!

Let's start

by making an

agrment.

?

The next step is to find

a the places where two

terms aren't in the natural

order—meaning the places

where two indexes have to

be switched for them to be

in an increasing order.

We gather a this

information into a

table like this.

Whoa.

Then we count

how many switches

we nd for

eachterm.

If the number is

even, we write the

term as positive. If

it is o, we write

it as negative.

Squze

We wi say that the right index

is in its natural order if

That is, indexes have to be in

an increasing order.

Switch

Switch

Paern 1

Paern 2

Paern 3

Paern 4

Paern 5

Paern 6

Paern 1

Paern 2

Permutations

of 1–2

Permutations

of 1–3

Coesponding term

in the determinant

Switches

Switches

and

and

and

and

and

and

and

and

and

and

Coesponding term

in the determinant

Rule

3

104 Chapter 4 More Matrices

Like this.

H...

Try comparing our earlier

determinant formulas with the

columns “Coesponding term in

the determinant” and “Sign.”

Ah!

Wow,

they're

the same!

Exactly, and that's

the third rule.

Number of

switch es

Number of

switch es

Sign

Sign

Sign

Sign

and

and

and

and

and

and

and

and

and

and

Paern 1

Paern 2

Paern 3

Paern 4

Paern 5

Paern 6

Paern 1

Paern 2

Permutations

of 1–2

Permutations

of 1–3

Coesponding term

in the determinant

Switches

Switches

Coesponding term

in the determinant

Coesponding term

in the determinant

Coesponding term

in the determinant

Calculating Determinants 105

These thr rules can

be used to find the

determinant of any

matrix.

Cl!

So, say we wanted

to calculate the

determinant of this

4×4 matrix:

Using this information,

we could calculate

the determinant if we

wanted to.

Agh!

Paern 1

Paern 2

Paern 3

Paern 4

Paern 5

Paern 6

Paern 7

Paern 8

Paern 9

Paern 10

Paern 11

Paern 12

Paern 13

Paern 14

Paern 15

Paern 16

Paern 17

Paern 18

Paern 19

Paern 20

Paern 21

Paern 22

Paern 23

Paern 24

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

2 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 1

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

3 & 2

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 1

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 2

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

4 & 3

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

4

4

4

4

4

4

a

11

a

11

a

11

a

11

a

11

a

11

a

12

a

12

a

12

a

12

a

12

a

12

a

13

a

13

a

13

a

13

a

13

a

13

a

14

a

14

a

14

a

14

a

14

a

14

a

22

a

22

a

23

a

23

a

24

a

24

a

21

a

21

a

23

a

23

a

24

a

24

a

21

a

21

a

22

a

22

a

24

a

24

a

21

a

21

a

22

a

22

a

23

a

23

a

33

a

34

a

32

a

34

a

32

a

33

a

33

a

34

a

31

a

34

a

31

a

33

a

32

a

34

a

31

a

34

a

31

a

32

a

32

a

33

a

31

a

33

a

31

a

32

a

44

a

43

a

44

a

42

a

43

a

42

a

44

a

43

a

44

a

41

a

43

a

41

a

44

a

42

a

44

a

41

a

42

a

41

a

43

a

42

a

43

a

41

a

42

a

41

2

2

3

3

4

4

1

1

3

3

4

4

1

1

2

2

4

4

1

1

2

2

3

3

3

4

2

4

2

3

3

4

1

4

1

3

2

4

1

4

1

2

2

3

1

3

1

2

4

3

4

2

3

2

4

3

4

1

3

1

4

2

4

1

2

1

3

2

3

1

2

1

0

1

1

2

2

3

1

2

2

3

3

4

2

3

3

4

4

5

3

4

4

5

5

6

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

+

−

−

+

Permutations

of 1–4

Switches

Sign

Num. of

switches

Coesponding term

in the determinant

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