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The Manga Guide to Linear Algebra by Ltd. Trend-Pro Co., Iroha Inoue, Shin Takahashi

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Calculating Determinants 101
You' have to learn
the thr rules of
determinants.
Thr
rules?
Yep, the terms in
the determinant
formula are formed
aording to
certainrules.
Take a closer lk at
the term indexes.
Pay special
aention 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 sm
a bit more
random.
Actuay, 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
Paern 1
Paern 2
Paern 1
Paern 2
Paern 3
Paern 4
Paern 5
Paern 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
agrment.
?
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
eachterm.
If the number is
even, we write the
term as positive. If
it is o, we write
it as negative.
Squze
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
Paern 1
Paern 2
Paern 3
Paern 4
Paern 5
Paern 6
Paern 1
Paern 2
Permutations
of 1–2
Permutations
of 1–3
Coesponding term
in the determinant
Switches
Switches
and
and
and
and
and
and
and
and
and
and
Coesponding term
in the determinant
Rule
3
104 Chapter 4 More Matrices
Like this.
H...
Try comparing our earlier
determinant formulas with the
columns “Coesponding 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
Paern 1
Paern 2
Paern 3
Paern 4
Paern 5
Paern 6
Paern 1
Paern 2
Permutations
of 1–2
Permutations
of 1–3
Coesponding term
in the determinant
Switches
Switches
Coesponding term
in the determinant
Coesponding term
in the determinant
Coesponding term
in the determinant
Calculating Determinants 105
These thr rules can
be used to find the
determinant of any
matrix.
Cl!
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!
Paern 1
Paern 2
Paern 3
Paern 4
Paern 5
Paern 6
Paern 7
Paern 8
Paern 9
Paern 10
Paern 11
Paern 12
Paern 13
Paern 14
Paern 15
Paern 16
Paern 17
Paern 18
Paern 19
Paern 20
Paern 21
Paern 22
Paern 23
Paern 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
Coesponding term
in the determinant

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