1
00:00:06,739 --> 00:00:07,530
ANA RITA PIRES: Hi.
2
00:00:07,530 --> 00:00:09,132
Welcome back to recitation.
3
00:00:09,132 --> 00:00:11,340
In lecture, you've been
learning about the properties
4
00:00:11,340 --> 00:00:12,600
of determinants.
5
00:00:12,600 --> 00:00:15,040
To remember, there were
three main properties,
6
00:00:15,040 --> 00:00:17,680
and then seven more that
fall out of those three.
7
00:00:17,680 --> 00:00:19,309
I'll tell you what
these three were.
8
00:00:19,309 --> 00:00:21,350
The first one was the
determinant of the identity
9
00:00:21,350 --> 00:00:23,850
matrix is always equal to 1.
10
00:00:23,850 --> 00:00:25,780
If you switch two
rows in a matrix,
11
00:00:25,780 --> 00:00:27,730
the determinant switches sign.
12
00:00:27,730 --> 00:00:30,770
And the determinant
is a function
13
00:00:30,770 --> 00:00:34,630
of each-- it's a linear
function of each row separately.
14
00:00:34,630 --> 00:00:36,780
And there's seven more.
15
00:00:36,780 --> 00:00:38,430
We'll use them here.
16
00:00:38,430 --> 00:00:41,630
Today's problem is about finding
the determinants of matrices
17
00:00:41,630 --> 00:00:43,380
by using these properties.
18
00:00:43,380 --> 00:00:46,640
So here you have four matrices.
19
00:00:46,640 --> 00:00:50,600
A has lots of 100's,
200's, and 300's numbers.
20
00:00:50,600 --> 00:00:54,180
B is called a
Vandermonde matrix.
21
00:00:54,180 --> 00:00:57,560
It has a very nice structure
with 1's, and then a, b, c;
22
00:00:57,560 --> 00:00:59,130
a squared, b squared, c squared.
23
00:00:59,130 --> 00:01:00,830
It can be bigger,
and you'll just
24
00:01:00,830 --> 00:01:04,750
have cubes and more
letters down here.
25
00:01:04,750 --> 00:01:07,380
C is given by the
product of these two,
26
00:01:07,380 --> 00:01:11,510
and D is this matrix.
27
00:01:11,510 --> 00:01:12,050
Good luck.
28
00:01:12,050 --> 00:01:14,669
Hit pause, work on them, and
when you're ready, come back
29
00:01:14,669 --> 00:01:15,960
and I'll show you how I did it.
30
00:01:25,290 --> 00:01:26,440
Did you get some?
31
00:01:26,440 --> 00:01:28,910
OK, let's do it.
32
00:01:28,910 --> 00:01:36,770
Starting with matrix A. I
have lots of big numbers.
33
00:01:41,400 --> 00:01:44,960
I suggest that we do a
little bit of elimination,
34
00:01:44,960 --> 00:01:47,250
because as you know,
doing elimination steps,
35
00:01:47,250 --> 00:01:49,870
except for permuting
rows, doesn't change
36
00:01:49,870 --> 00:01:52,100
the determinant of the matrix.
37
00:01:52,100 --> 00:02:04,660
So let's do determinant of A
is equal to-- 101, 201, 301.
38
00:02:04,660 --> 00:02:08,580
Then if I subtract off the
first row from the second one,
39
00:02:08,580 --> 00:02:12,000
I'll get 1, 1, 1, which
is very convenient.
40
00:02:12,000 --> 00:02:15,490
And actually, if you
subtract the second row
41
00:02:15,490 --> 00:02:21,480
from the third one,
you'll get 1, 1, 1.
42
00:02:21,480 --> 00:02:23,690
Here's the property
of the determinant:
43
00:02:23,690 --> 00:02:25,990
if you have two equal
rows on your matrix,
44
00:02:25,990 --> 00:02:28,105
the determinant is
automatically equal to zero.
45
00:02:30,750 --> 00:02:31,310
All right.
46
00:02:31,310 --> 00:02:32,420
All done with one of them.
47
00:02:32,420 --> 00:02:35,660
Let's work on the second one.
48
00:02:35,660 --> 00:02:42,940
The determinant of B. Well,
let's try elimination again.
49
00:02:42,940 --> 00:02:47,310
1, a, a squared.
50
00:02:47,310 --> 00:02:53,700
1 minus 1 is 0, b minus a,
and b squared minus a squared.
51
00:02:53,700 --> 00:02:56,060
b squared minus a squared--
let me factor that
52
00:02:56,060 --> 00:03:01,420
into b minus a, b plus
a, which will be very
53
00:03:01,420 --> 00:03:03,440
convenient in the next step.
54
00:03:03,440 --> 00:03:06,290
And then I'm going to
subtract the first row again
55
00:03:06,290 --> 00:03:07,935
from the third one.
56
00:03:07,935 --> 00:03:10,430
I'll get 0, c minus a.
57
00:03:10,430 --> 00:03:13,550
And again, I'll get c
squared minus a squared, c
58
00:03:13,550 --> 00:03:16,742
minus a, c plus a.
59
00:03:23,350 --> 00:03:30,680
Let's use that third property
that was the determinant
60
00:03:30,680 --> 00:03:34,270
is linear on each
row separately.
61
00:03:34,270 --> 00:03:38,320
So what I'm going to do is,
see this factor of b minus a?
62
00:03:38,320 --> 00:03:41,170
It shows up in every
entry of this row.
63
00:03:41,170 --> 00:03:45,900
Well it's a 0, so it's the
zero multiple of b minus a.
64
00:03:45,900 --> 00:03:48,620
So I'm going to pull out
this factor of b minus a,
65
00:03:48,620 --> 00:03:53,710
and this row is going to
become 0, 1, b plus a.
66
00:03:53,710 --> 00:03:57,100
I will also, in the same
step, do the same thing
67
00:03:57,100 --> 00:03:58,230
with the third row.
68
00:03:58,230 --> 00:04:02,110
I'll pull out a
factor of c minus a,
69
00:04:02,110 --> 00:04:08,260
and it will become
0, 1, c plus a.
70
00:04:08,260 --> 00:04:11,540
Here's one factor
from the second row.
71
00:04:11,540 --> 00:04:14,640
Another factor
from the third row.
72
00:04:14,640 --> 00:04:18,420
And 1, a, a squared.
73
00:04:18,420 --> 00:04:22,200
0, 1, b plus a.
74
00:04:22,200 --> 00:04:25,410
0, 1, c plus a.
75
00:04:28,460 --> 00:04:29,760
Now what?
76
00:04:29,760 --> 00:04:32,250
Well remember,
you know how to do
77
00:04:32,250 --> 00:04:34,670
the determinant of upper
triangular matrices
78
00:04:34,670 --> 00:04:37,760
because all that you do
is multiply the pivots.
79
00:04:37,760 --> 00:04:40,610
This is almost upper triangular,
except there's a 1 over here.
80
00:04:40,610 --> 00:04:42,410
So let's do another
elimination step.
81
00:04:45,250 --> 00:04:50,170
b minus a, c minus a.
82
00:04:50,170 --> 00:05:00,850
1, a, a squared; 0, 1, b
plus a; 0, 0, c plus a minus
83
00:05:00,850 --> 00:05:06,400
b plus a is c minus b.
84
00:05:06,400 --> 00:05:10,540
So the determinant of this
matrix is now 1 times 1 times
85
00:05:10,540 --> 00:05:12,390
c minus b.
86
00:05:12,390 --> 00:05:20,670
So we get b minus a,
c minus a, c minus b,
87
00:05:20,670 --> 00:05:23,120
which has a really nice formula.
88
00:05:23,120 --> 00:05:25,540
This is called the
Vandermonde determinant.
89
00:05:25,540 --> 00:05:28,430
It's always like this, even if
your matrix is bigger than 3
90
00:05:28,430 --> 00:05:32,580
by 3, if it's 4 by 4, or
5 by 5 and so on, you just
91
00:05:32,580 --> 00:05:34,770
have more differences
of all the letters
92
00:05:34,770 --> 00:05:37,870
that show up in your matrix.
93
00:05:37,870 --> 00:05:41,190
On to matrix number 3.
94
00:05:41,190 --> 00:05:53,990
C equals [1, 2, 3] [1, -4, 5].
95
00:05:53,990 --> 00:05:58,010
How did you get the
determinant for this one?
96
00:05:58,010 --> 00:06:02,820
Well remember, this
is a rank one matrix
97
00:06:02,820 --> 00:06:05,900
because it's a column
vector times a row vector.
98
00:06:05,900 --> 00:06:08,850
So if you write out what
the matrix is it'll be a 3
99
00:06:08,850 --> 00:06:13,140
by 3 matrix where we can
think about it this way.
100
00:06:13,140 --> 00:06:17,210
The first row will be
1 times [1, -4, 5].
101
00:06:17,210 --> 00:06:20,260
The second row will be
2 times those numbers.
102
00:06:20,260 --> 00:06:23,870
And the third row will be
three times those numbers.
103
00:06:23,870 --> 00:06:30,960
So all the rows are going
to be linearly dependent,
104
00:06:30,960 --> 00:06:34,492
or another way of saying
the matrix is singular.
105
00:06:34,492 --> 00:06:36,560
When the matrix is
singular, the determinant
106
00:06:36,560 --> 00:06:37,750
is always equal to 0.
107
00:06:37,750 --> 00:06:40,580
That's also one
of the properties.
108
00:06:40,580 --> 00:06:42,490
C is 0.
109
00:06:45,930 --> 00:06:48,240
Onto the next matrix.
110
00:06:48,240 --> 00:06:49,690
Last one.
111
00:06:49,690 --> 00:06:56,660
D is equal to 0, 0, 0.
112
00:06:56,660 --> 00:07:06,750
1, -1, 3, -3, 4, -4.
113
00:07:06,750 --> 00:07:10,800
Maybe you didn't get that from
just looking at the matrix
114
00:07:10,800 --> 00:07:13,105
the first time, but did you
see how I wrote it down?
115
00:07:13,105 --> 00:07:15,180
It has 0's down the diagonal.
116
00:07:15,180 --> 00:07:20,130
And then for each entry,
I have minus that entry.
117
00:07:20,130 --> 00:07:22,660
That means that this
matrix is skew symmetric.
118
00:07:22,660 --> 00:07:25,940
What that means is if
you do D transpose,
119
00:07:25,940 --> 00:07:27,590
it will not be
equal to D, but it
120
00:07:27,590 --> 00:07:37,430
will be equal to minus D. D
transpose is equal to minus D.
121
00:07:37,430 --> 00:07:39,890
Now, what does this give
us for a determinant?
122
00:07:39,890 --> 00:07:43,640
Well if these two matrices
are the same matrix,
123
00:07:43,640 --> 00:07:46,830
this determinant is equal
to that determinant.
124
00:07:46,830 --> 00:07:49,750
One of the properties
is that the determinant
125
00:07:49,750 --> 00:07:53,780
of a transpose of the matrix
is equal to the determinant
126
00:07:53,780 --> 00:07:56,100
of the original matrix.
127
00:07:56,100 --> 00:07:57,460
How about this side?
128
00:07:57,460 --> 00:08:07,350
Well, first temptation
would be to just write that.
129
00:08:07,350 --> 00:08:09,330
Is that always true?
130
00:08:09,330 --> 00:08:09,870
No.
131
00:08:09,870 --> 00:08:14,120
The determinant is linear
on each row separately.
132
00:08:14,120 --> 00:08:16,650
That means that you can't
pull out the factor that
133
00:08:16,650 --> 00:08:18,010
is multiplying the matrix.
134
00:08:18,010 --> 00:08:21,330
You have to pull it out once
for each row of the matrix.
135
00:08:21,330 --> 00:08:31,560
So what I should
have written was
136
00:08:31,560 --> 00:08:34,179
-1, that's my factor, minus.
137
00:08:34,179 --> 00:08:35,370
How many rows do I have?
138
00:08:35,370 --> 00:08:37,490
One, two, three.
139
00:08:37,490 --> 00:08:39,690
Pull it out once for each row.
140
00:08:39,690 --> 00:08:44,990
Times the determinant of D. Well
fortunately, -1 to the third
141
00:08:44,990 --> 00:08:48,370
is simply equal to -1.
142
00:08:48,370 --> 00:08:50,860
So here we go.
143
00:08:50,860 --> 00:08:54,490
It was correct, in fact.
144
00:08:54,490 --> 00:08:57,190
We have determinant of D is
equal to minus determinant
145
00:08:57,190 --> 00:08:59,380
of D. What is the
only number that
146
00:08:59,380 --> 00:09:01,970
is equal to minus that number?
147
00:09:01,970 --> 00:09:03,580
0.
148
00:09:03,580 --> 00:09:08,991
Determinant of D is
equal to 0 again.
149
00:09:08,991 --> 00:09:11,830
Let me ask you
one last question.
150
00:09:11,830 --> 00:09:14,330
Is it true that all skew
symmetric matrices have
151
00:09:14,330 --> 00:09:16,355
the determinant equal to 0?
152
00:09:16,355 --> 00:09:18,190
It was true for this one.
153
00:09:18,190 --> 00:09:21,010
Is it true in every case?
154
00:09:21,010 --> 00:09:24,490
Well, the key factor
here was that I
155
00:09:24,490 --> 00:09:29,260
had -1 to the third power
and I got a minus sign here,
156
00:09:29,260 --> 00:09:32,215
determinant of D is equal
to minus determinant of D.
157
00:09:32,215 --> 00:09:34,920
What if this number had
been an even number?
158
00:09:34,920 --> 00:09:37,500
Then I would just have
the determinant of D
159
00:09:37,500 --> 00:09:42,350
is equal minus 1 to an
even number, D. That's 1.
160
00:09:42,350 --> 00:09:46,420
So I would have determinant of
D is equal to determinant of D.
161
00:09:46,420 --> 00:09:48,380
There's nothing I can
say about that number.
162
00:09:48,380 --> 00:09:51,505
It can be your favorite
number, not necessarily 0.
163
00:09:51,505 --> 00:09:52,510
All right.
164
00:09:52,510 --> 00:09:53,500
We're done for today.
165
00:09:53,500 --> 00:09:54,848
Thank you.