00:00Hi guys, so in this
00:01lesson we're going to find
00:02the determinant of matrix and
00:04the inverse of a matrix.
00:06So firstly, what is the
00:08inverse of a matrix? Well,
00:11we imagine we have a
00:14matrix A. Well, the inverse
00:17of A is the matrix
00:19you multiply it by to
00:21get the identity matrix. So
00:24what do I multiply A
00:25by to get the identity
00:26matrix? Well, it's
00:28this inverse of A. And
00:32the way we get it
00:33is, quite simply, this, the
00:35inverse of A is 1
00:36over the determinant of A
00:38times this where you basically
00:40swap A and D and
00:41change the signs of B
00:42and C, and the determinant
00:44of A is AD minus
00:45BC. Now you may say,
00:48well hang on, no, that
00:48makes any sense. What is
00:50this? Well, I'm not going
00:53to actually prove this to
00:55you, although it's
00:56It's actually not that difficult
00:57to do. And I'll just
00:58show you how you can
00:59do it yourself, but I'm
01:00not going to do it
01:01in this video. If you
01:02get a B, C, D,
01:04get this matrix A, B,
01:05C, D, and multiply it
01:07by x, now now x,
01:09w, x, y, and z.
01:13And you multiply this by
01:14this, using our multiplication rules
01:17from matrices. And you set
01:19it equal to the identity
01:21matrix, which is this.
01:24When you solve for wxy
01:26and z, you end up
01:28getting this and that's it.
01:31And if you try, it's
01:32actually a nice exercise to
01:33do that. You get a
01:34clearer understanding of why this
01:35is the inverse matrix. Now,
01:41few things, it has to
01:42be a square matrix. You
01:43can only get the inverse
01:45of a square matrix. You
01:46can only get the determinant
01:48of a square matrix in
01:50this
01:52Applications course you need to
01:55know how to get the
01:56inverse of a 2 by
01:572 matrix by hand You
02:00calculator can do it, but
02:02all they need to do
02:03is put a letter in
02:04instead of that 2 and
02:06You calculator can do it.
02:07So you need to be
02:09able to do it by
02:10hand for a 2 by
02:112 a 3 by 3
02:13which is the example I'm
02:14going to show here You're
02:16not expected to do it
02:17by hand. You just have
02:17to do it with the
02:18calculator. So I'm going to
02:19do three examples
02:20The first one is by
02:22hand. The second one, I'm
02:23going to show you actually
02:24doesn't have an inverse and
02:27the third one, I'll get
02:28to this bit when I
02:29get to the second part
02:30here and the third one
02:32I'll do it using the
02:32calculator. Okay, so I'm going
02:34to call this a, b,
02:36and c just like this.
02:38So if this is a,
02:39this is pretty straightforward guys.
02:41A is three, two, negative
02:43one, one. Actually, I don't
02:45need to write that out
02:46again. The first thing I
02:47do
02:48is I get the determinant
02:51of a, debt, a determinant
02:54of a. Now you may
02:54be asking why is it
02:55called the determinant? Well this
02:58is actually quite an important
02:59property of square matrices and
03:02we'll be learning a bit
03:03more about it in future
03:05lessons for now. Just accept
03:07that the determinant is a,
03:09d minus b, c. So
03:10this tells us, minus this
03:11tells us. So three times
03:13one, three times one, which
03:15is three.
03:16minus 2 times negative 1
03:19is negative 2, which is
03:21equal to 5. So the
03:24determinant of a is 5,
03:26which means a minus 1,
03:29the inverse matrix of a
03:31is equal to 1 over
03:335, because it's 1 over
03:35the determinant of the determinant
03:36of a times. I'm going
03:39to swap these two and
03:42change the signs of these
03:43two. So swap this because
03:44one and three and change
03:47the signs of these becomes
03:48one and negative two and
03:50that's it. That's the inverse
03:52of A. I could write
03:54it as one -fifth, negative
03:58two -fifths, one -fifth, three
04:01-fifths. If I wanted, but
04:05it doesn't say to do
04:06so, so this is absolutely
04:09fine. Second one.
04:12Okay. Let's get the determinant
04:15of b. Same thing. The
04:16determinant of b is equal
04:18to three times one minus
04:20three times one, which is
04:24zero. Now, if the determinant
04:26of a matrix is zero,
04:27we've got a problem because
04:30the determinant, if the determinant
04:33is zero, the inverse is
04:34going to be one over
04:35zero times this. So we've
04:39got a big problem because
04:40we're dividing my zero.
04:40So I'm going to refer
04:42you to this little bit
04:43of red writing written up
04:45here. If the determinant of
04:47A is zero, then A
04:49is singular. So it's a
04:51singular matrix, which means it
04:52does not have an inverse.
04:54That's an important word of
04:55singular. If the determinant of
04:57A is not equal to
04:58zero like this one, then
05:00it is non singular. Or
05:02in other words, you have
05:04to be familiar with this
05:05invertible. It is an invertible
05:07matrix, which means it has
05:08an inverse.
05:08So a minus one exists
05:10here does not exist here.
05:12It does exist. So therefore
05:14B minus one does not
05:21exist and finally part C
05:25find the inverse of C.
05:29So let's see minus one.
05:32Okay, so we're going to
05:35do this with
05:36are trusty GDC. So you
05:42can actually do that by
05:43hand tonight when I taught
05:45further maths aloes. You actually
05:48did have to do that
05:48by hand in the IB.
05:50Luckily you don't have to
05:51do it. It can actually
05:52be quite annoying. So what
05:55I'm going to do is
05:57menu matrix and vector create.
06:01I'm going to create a
06:02matrix we've done this before.
06:04So a number of roses
06:04It's three number of columns
06:07is three. I literally just
06:09put in the numbers one,
06:10three, one. This is one,
06:14four. Zero, two, negative one.
06:22Zero. And I'm gonna store
06:25this, remember we did this
06:26before, let's store this as
06:28C. Let's call this C.
06:33Now to get the inverse,
06:35I literally just do C
06:36to the power of negative
06:391. Easy, done, that's it.
06:42The inverse of C is
06:45negative 1, 1, sorry, negative
06:471, 1, 1, negative 2,
06:522, 1, and 8, negative
06:577, negative 4, 8, negative
07:01If seven negative four, close
07:04these brackets properly. One more
07:08time. There we go. And
07:10that's the inverse. Just the
07:12final thing I'm going to
07:13show you is if they
07:15just ask you to get
07:16the determinant, I can say
07:18menu, matrix and vector and
07:24just click your determinant. So
07:26I do debt, C, press
07:28enter and the
07:29determinant of C is negative
07:311. You can do that
07:32for a just a test
07:34test your answer. Okay, that's
07:37it. That's the lesson. Hopefully,
07:39now, you know what an
07:40inverse matrix is. The inverse
07:43of a matrix, you know
07:44how to find the determinant
07:45of a matrix for a
07:472 by 2 or any
07:49other matrix using the calculator.
07:51And you know how to
07:52find the inverse of a
07:53matrix for, again, a 2
07:55by 2 by hand and
07:56any other one.
07:57using the calculator and you
07:58know that the determinant is
08:00zero, it does not have
08:02an inverse and it is
08:03singular. If it does have
08:05an inverse, it is invertible
08:08or non -singular. Okay, that's
08:11it. I will see you
08:13in the next lesson.