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00:00:00In Chapter 4, we will talk about Inverse Matrix.
00:00:05Inverse Matrix is A-Inverse.
00:00:14A-Inverse looks like a square root of 1.
00:00:20A-Inverse can be thought of in many ways.
00:00:24It can be thought of in many ways.
00:00:31Today, we will talk about how to do A-Inverse Matrix.
00:00:37The first thing is using adjoints.
00:00:40You can use an Adjoint Matrix.
00:00:44But if you use an Adjoint Matrix, it can't be done more than 3 x 3.
00:00:48Otherwise, you can't do it.
00:00:50You have to think about the cofactors, right?
00:00:52It's quite a lot.
00:00:54The second type is the row operation.
00:01:00It's very fast. You can use it for everything.
00:01:02A inverse has a rule that if the matrix A has an inverse, it's a non-singular matrix.
00:01:24If it has an inverse, it's a non-singular matrix.
00:01:28If it has an inverse, it's a non-singular matrix.
00:01:38If it has an inverse, it's a non-singular matrix.
00:01:54If A has an inverse, it's a non-singular matrix.
00:02:04If A has an inverse, it's a non-singular matrix.
00:02:14Now, let's talk about the adjoint matrix.
00:02:26The first thing is to think about the inverse matrix by using the adjoint matrix.
00:02:32The adjoint matrix is the matrix of the cofactors.
00:02:46It's very easy to think about A inverse.
00:02:48Just add one part of det A.
00:02:50But as I said, to think about A inverse, det A doesn't have to be 0.
00:02:56It has to be a non-singular matrix.
00:03:08Just add one part of det A.
00:03:12Then, we get A inverse in the adjoint matrix.
00:03:24For example, this one.
00:03:26They told us to find A inverse of this one.
00:03:28But this one, we have already thought about its cofactors.
00:03:32Let me go back and find the same one.
00:03:34It should be the same.
00:03:36Which one?
00:03:38Is it this one?
00:03:40I don't think so.
00:03:48It should be this one.
00:03:50It should be this one.
00:03:52It's the same, right?
00:03:54Let's try to find the same one.
00:03:56It's the same, right?
00:03:58From the 27th, we have already thought about the adjoint matrix.
00:04:06Let's try to find the adjoint matrix again.
00:04:10From the 27th, we have already thought about the adjoint matrix.
00:04:18It should be this one.
00:04:22Otherwise, it's too long.
00:04:24From the 27th, we have already thought about the adjoint matrix.
00:04:30It should be this one.
00:04:34From the 27th, we have already thought about the adjoint matrix.
00:04:38It should be the same.
00:04:48From the 27th, we have already thought about the adjoint matrix.
00:04:56But we have to do it again.
00:04:58It's too long, right?
00:05:00We have found it.
00:05:02Our job is to think about the adjoint matrix.
00:05:10The adjoint matrix must not be zero.
00:05:18Let's add some values.
00:05:30Let's add some values.
00:05:32Then multiply it by 8.
00:05:38Then multiply it by 3.
00:05:42Then multiply it by 0.
00:05:44Then multiply it by 11.
00:05:46Then multiply it by 36.
00:05:52Then multiply it by 0.
00:05:54Then multiply it by 40.
00:06:00Then multiply it by 40.
00:06:02We get 51.
00:06:06We get 51.
00:06:12We get 51.
00:06:16We can find A inverse.
00:06:20But the step before we can find the adjoint is 5 pages.
00:06:24But the step before we can find the adjoint is 5 pages.
00:06:28So, A inverse.
00:06:30I forgot to tell you that A is not zero.
00:06:34A is not zero.
00:06:36So, it's a non-singular matrix.
00:06:42When it's a non-singular matrix, we can find A inverse.
00:06:52When it's a non-singular matrix, we can find A inverse.
00:06:54A inverse is equal to
00:06:561 over det A
00:07:041 over det A
00:07:06is equal to adjoint A
00:07:08is equal to adjoint A
00:07:10transpose
00:07:121 over det A
00:07:14is equal to adjoint A
00:07:16transpose
00:07:181 over det A
00:07:20is equal to adjoint A
00:07:22transpose
00:07:241 over det A
00:07:26is equal to a factor A
00:07:28transpose
00:07:301 over det A
00:07:32is equal to a factor A
00:07:34If we multiply it by 51.
00:07:36If we multiply it by 51.
00:07:38Then it becomes A inverse.
00:07:42you
00:09:12can
00:09:26use
00:09:48low
00:10:12operation
00:10:32in
00:11:22the
00:12:18next
00:13:13time
00:14:08This is a non-singular matrix, which has a value less than 0, so it is a non-singular matrix.
00:14:39This is a non-singular matrix, which can find A-inverse.
00:14:49This is one of the rules that can help you find A-inverse.
00:14:59You can test it or not, it's up to you.
00:15:03If the matrix is very large, you don't need to test it.
00:15:08If the matrix is very large, you don't need to test it.
00:15:13Note that you can test it or not.
00:15:20If the matrix is 2x2 or 3x3, you can test it.
00:15:27So you don't waste your time.
00:15:29If there is no inverse, you can use this reason to prove that there is no inverse.
00:15:33So you don't waste your time.
00:15:39I will start with elementary row operation.
00:15:53Find A-inverse.
00:15:54Find A-inverse.
00:16:03Find A-inverse.
00:16:07Find A-inverse.
00:16:14Find A-inverse.
00:16:15So, N is 3, and the matrix A is 1, 2, 1, minus 1, minus 1, 1, 0, 1, 3, and add them to the matrix I, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
00:16:450, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:17:150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:17:450, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:18:150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:18:450, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:19:150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:19:450, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:20:150, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
00:20:450, 0, 0, 0, 0
00:21:150, 0, 0, 0, 0, 0, 0, 0, 0
00:21:400, minus 2, minus 2
00:21:470, minus 0, is 0
00:21:52Is it straight?
00:21:54It's straight, ok
00:21:56Pull, pull, pull, pull, pull
00:21:58It's here, see?
00:22:00It's complete, click, click, click, click
00:22:01Here, 1 is here, 1 is complete
00:22:03Make this up here become 0
00:22:05Make this up here become 0
00:22:07Then, change to line R1 again
00:22:09Change to line R2
00:22:11By adding line R3
00:22:13Plus R3
00:22:15Minus 2R3
00:22:19This will become 0
00:22:21And the last, what is this side?
00:22:23It's A-invert
00:22:27You don't have to think about this
00:22:29It will be like this
00:22:31See?
00:22:33If we follow the plan
00:22:35It will become a single matrix
00:22:37It will become 1, 1, 1, 1
00:22:39Click, click, click, click, click
00:22:43It will be like this
00:22:45There are only numbers left
00:22:47The top row, this one doesn't change
00:22:49There are only numbers left
00:22:51Here
00:23:01R1 plus 3, 5, 3
00:23:03R1 plus 3, R3
00:23:05R1 is minus 1
00:23:07Minus 1, minus 3
00:23:09Minus 4
00:23:13Minus 2, minus 3
00:23:15Minus 5
00:23:21And 0 minus
00:23:230 plus 3
00:23:250 plus 3 is 3
00:23:27In line R2
00:23:291
00:23:31Plus 2 is 3
00:23:331
00:23:35Plus 2 is 3
00:23:39And 0
00:23:41Minus 2
00:23:43Minus 2
00:23:45Is it correct?
00:23:49And what is this?
00:23:51This will become a matrix
00:23:53R3 is on this side
00:23:55See?
00:23:57R3 is on the left side
00:23:59And this matrix B
00:24:01Is A inverse
00:24:05We can write it like this
00:24:07We can write R3 plus A inverse
00:24:09Can you remember?
00:24:11It will become A inverse
00:24:13This is the matrix inverse
00:24:15Therefore, we will answer
00:24:17A inverse equals
00:24:19Minus 4, minus 5
00:24:21Minus 3, minus 3
00:24:23Minus 2, minus 1
00:24:25Minus 1
00:24:27It is more convenient
00:24:29than using
00:24:31cofactor
00:24:33Adjoint is slower
00:24:35This is faster
00:24:41Low operation
00:24:43They don't tell us how to do it
00:24:45We choose it
00:24:49Unless
00:24:51they tell us
00:24:53We can't do it
00:24:55We can't choose
00:24:57cofactor by adjoint
00:24:59Most of the time, they won't tell us
00:25:01Because they want us to do
00:25:03low operation
00:25:05Adjoint doesn't care
00:25:07much
00:25:09They only care about low operation
00:25:11What?
00:25:13Yes
00:25:15Just think it right
00:25:17It is a faster way
00:25:19It is made for this
00:25:21It is made to solve the problem
00:25:23That's all
00:25:25Next
00:25:27Let's try
00:25:29R3 plus A inverse
00:25:31Why do I write it?
00:25:33I can write A inverse
00:25:35Low operation is the fastest
00:25:39Let's try
00:25:41It is rejected
00:25:43Isn't it?
00:25:45It is rejected
00:25:53Let's think about A inverse
00:25:55Does it have A inverse?
00:25:57It doesn't have
00:26:05Let's think about A inverse
00:26:07If it is 3x3
00:26:09It won't waste time
00:26:11If it doesn't have
00:26:13It is rejected
00:26:238
00:26:250
00:26:2920
00:26:350
00:26:394
00:26:432
00:26:47It should have
00:26:4912 minus 8
00:26:518
00:26:534
00:26:55A is not equal to 0
00:26:57It is a non-singular matrix
00:27:05If it is a non-singular matrix
00:27:07We can find A inverse
00:27:15It is easier
00:27:17to find A inverse
00:27:21Let's try
00:27:33It is a non-singular matrix
00:27:39We can find A inverse
00:27:41Now, we use elementary low operation
00:28:21I think you haven't finished the lesson yet.
00:28:25The lesson I'm talking about.
00:28:27Do it yourself.
00:28:31Add I3.
00:28:33See? Add 1, 1, 1.
00:28:350, 0, 0.
00:28:37Roll operation to make this side a matrix.
00:28:39A singular matrix.
00:28:41Take singular from this side
00:28:43and make it return to this side instead.
00:28:45How to make it beautiful?
00:28:49Is it beautiful to swap first?
00:28:51Or not to swap?
00:28:53It's up to you.
00:28:55I'll swap.
00:28:57I'll swap R2 and R1.
00:29:01R1, R2.
00:29:03Always R2.
00:29:15I don't like R2.
00:29:29I'll swap R1 and R2.
00:29:33I'll swap R1 and R2.
00:29:35I'll swap R1 and R2.
00:29:37I'll swap R1 and R2.
00:29:39I'll swap R1 and R2.
00:30:09I'll swap R1 and R2.
00:30:11I'll switch R1 and R2.
00:30:13I'll swap R1 and R2.
00:30:15I'll switch R1 and R2.
00:30:17I'll make this side a T grid.
00:30:29This way I can make this side whole.
00:30:31This side here is the mean.
00:30:33Do it with me.
00:30:37I'll tell you Ben soon. But I need me new now. Hey man, I'm high come on Ben soon. I need a mean in La
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00:35:29it will be changed to 2, change it to 0
00:35:341 plus 4 is 5
00:35:39this one
00:35:445 minus 1 is 4
00:35:49right? this one is 1 as before
00:35:546 minus 1 is 1 and 0
00:35:59ok
00:36:045 minus 4, can you do it?
00:36:09change it to 2, 3
00:36:14you don't have to change it, you can use 5 minus 4
00:36:19change it to 0
00:36:24it's up to you
00:36:29it's up to you
00:36:34021, 001
00:36:39054, 001
00:36:44001, 110
00:36:49change it to 2
00:36:541252, change it to 2
00:37:14ok
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