Alrighty then.
So moving on up to three by three and larger matrices, but we are going to do a 3x3 in
our example.
How do you find a multiplicative inverse of an NxN matrix where n is greater than two.
Instead of a 2x2 matrix, 3x3..4x4..5x5..there is no really easy process, at least if you
are doing this math by hand, for finding that inverse matrix.
We are going to form an augmented matrix in the form of A with that augmented line and
on the right hand side put the identity matrix.
Then we are going to perform some row operations, some Gaussian Elimination process if you will,
on this augmented matrix.
I want to make the left side of this augmented matrix the identity matrix with that diagonal
of ones and everything else zeros.
What is on the right hand side of that augmented matrix is hopefully going to be the inverse.
B, this new matrix that we are going to get on the right side of our augmented matrix
is the inverse of matrix A or the inverse of A. But, you want to make sure that answer
so make sure that you check still that matrix A times what we believe is the matrix of A
equals the identity matrix.
If you change the order of that multiplication you will again get that identity matrix.
That will be your check to see if your inverse matrix is just that.
We are going to find the inverse of this matrix A, this 3x3 matrix.
We are first going to write that augmented matrix.
We are going to say that we have got 1, -2, 2, 1, 0, -2, -2, 1, 3 augment that with the
identity matrix I sub three on the right hand side.
It is very easy to make mistakes when you are working with these matrices so I am going
to be very meticulous with my work here an not make any errors.
I am going to check along the way to make sure that I don't make a sign error or a basic
arithmetic error along this process and then get this board full of work and realize that
there is a mistake somewhere, because then I will have to reshoot this whole scene.
When we do our row operations to get this left hand side to be the identity matrix,
you should have already studied this when you were solving systems of equations using
Gaussian Elimination process or the row operations process.
We are going to work on getting this bottom left hand triangle to be all zeros.
Then we are going to get this last element in the third row to be that one that need
of the identity matrix.
Then work our way up into the upper right hand corner and get all those elements to
be zero so we have that diagonal row of ones with all of the elements on the left hand
side being equal to zero.
I can see here that I have a negative two and a positive two, so I am already set up
to go ahead and add row two with row three and get this first element in the third row
to be equal to zero.
We are going to come down here and we are going to have 1, 1, -2, 1, 0, 0.
Next row -2, 0, 1, 0, 1, 0.
Now I am going to for this third row, I am going to take row two from my, this is how
I show the work, from my previous step here...
I am going to take row two and add it with row three.
Just to make sure that if I do make an error through my work, maybe with all this notation
I can see what the heck I did and find that error.
Sometimes it is so hard to find your error when you are doing Gaussian Elimination it
is almost better to crumple up your paper and start over again, there is going to be
so much arithmetic.
Negative two plus two is equal to zero.
Zero plus negative two is negative two.
One and three is four.
Then we have zero, adding these two we have one, and then zero and one is equal to one.
Now let's see here, what else can we do.
I need this value now, or I could...
It doesn't really....
Actually yes, I don't really want to work on making that a zero right now, not with
this first row because this first row has an element here of one and if I try combining...if
I want to take the first row and multiply it by two and add it with the third row then,
yeah, I am going to do two plus zero and remove that zero that I have already created there.
Let's work on making this equal to zero by working with the first row.
If that is negative two, if I can make that positive two, positive two and negative two
is going to add to an answer of zero.
For my next line of work, I am going to leave this bottom one the same.
I am going to keep the top row the same as well.
To create a zero here I am going to take, let's see, I need that again...if that is
negative two I want this first element to be positive two.
I am going to do two times row one and add that, now two times row one again... two times
one is two plus negative two is zero...so i can leave that second row alone.
I am going to do 2R1 plus R2.
Again with the way that I show my scratch work, I am going to go up to the previous
line and multiply every element in row one by two.
2, 2, -4, 2, 0, and 0.
Negative two plus, excuse me, positive two plus negative two or 2 - 2 is zero.
Then we have two plus zero is two.
Now we have negative four plus one is negative three.
Two plus zero is two.
Zero and one is one.
Zero and zero is zero.
Now, this is nice.
Now I need to get the second element in my third row to be zero.
I need that bottom left hand corner to be all zeros.
Now that I have got the two 1st elements in row two and row three, those are both equal
to zero so I can safely work with row two and row three to create a zero here without
being afraid of losing my zero.
I cannot now work with row one and row three together because I don't want to lose that
zero.
These are already opposite numbers, so really I can just right now add row two and row three
and be in a good position.
So, we are going to bring this up here and we are going to not touch row one.
1, 1, -2, 1, 0, 0 We are going to leave row two alone, just doing one step at a time.
Now for my new third row I want to just add the second and the third rows.
Zero and zero is zero.
Two plus negative two is zero.
Negative three plus four is one.
We have two, two, and then finally one.
I unfortunately suffer from the same affliction as some of my kids where I find it very easy
to make silly mistake and not catch them.
We are good so far.
Now that I have got my bottom left hand corner to be all zeros.
Let's start work on creating those zeros in the upper right hand corner.
Ok.
I want to work my way up and I don't want to lose these zeros either.
I don't know why I am having so much trouble with static today.
I don't want to try to create a zero here using the first row because I am going to
lose that zero so I want to work my way up.
If I a make that a three and I add it with the negative three, I am going to get that
zero that I need.
So, are going to do on the next step of work here, we are going to leave that first row
alone again.
Here is my last row.
I want to add, let's just stand over here, I want to add the second row and the third
row and get that to be zero, so that needs to be positive three to add with negative
three and get zero.
So, we are going to do, let's see here, about row two
plus three times row three.
I am going to come up here and I am going to take three and multiply it to all of these.
That is zero, zero, one times three is three, two times three is six, six, and one.
Let's add those up.
We have row two plus three times row three.
Zero and zero is zero.
We have two and zero is going to be still two.
We have negative three and three which is going to give me zero.
We have two and six which is eight.
One and six which is seven, and zero and one which is one.
Excellent, now I want to make that negative two equal to zero.
I am going to do that by, if I can just make that equal to two which of course I can by
multiplying by two and then adding it with row one, that is going to be zero and I will
almost be done.
So we are going to take row one and it is already negative so I am going to add it with
two times row three.
Everything else is going to stay the same so let's just get that down.
The last row is 0, 0, 1, 2, 2, and 1.
The second, we are leaving that alone right now.
You can see there is nothing out here.
Alright, I want to take the first row and add it with two times row three.
I am going to go in here and do this.
Two times row three, two times two, times two, times two, times two, times two, and
times two.
Now add row one plus two times row three.
One plus zero is going to be one.
One plus zero is one.
Negative two plus two is equal to zero.
We have got one plus four is five, here is four, and two.
Excellent.
Now, I have got my one down here that I need.
I have got a two, I can divide by two but I don't want to create any fractions before
I absolutely need to.
So, we are going to make that second element in the first row equal to zero by making it
equal to negative two temporally so that I can add it with the positive two and get zero.
So, on this next step of work we are going to change that first row again.
It is going to be, we need that to be negative two so that I can add it with positive two
and get zero.
We are going to do negative two times row one and add it with row two.
Everything else is going to stay the same.
Let's get that copied down.
The last row is 0, 0, 1, 2, 2, 1.
The second row is going to stay the same for now 0, 2, 0, 8, 7, 1.
I don't know what the voice is about, just to lighten up the mood a little bit.
Negative two times row one is going to be negative two, negative two, negative two times
zero is zero, then we have negative ten, and negative two times four is negative eight,
and negative four.
Adding those up we get negative two plus zero is negative two.
We have negative two plus positive two is that zero that we need building the lower
corner and now the upper corner to be all zeros.
Of course zero plus zero is zero.
Negative ten and eight is negative two.
Negative eight and seven is negative one.
Negative four and one is negative three.
Let's just make sure that I am at a good spot right now.
Ok, so now at this point I am going to get my red pen out.
You know, do I shoot the video and make it look like the teacher never makes any mistake.
If you watch a lot of my videos, you know that is not the case with me especially.
Or, do now I take this as an opportunity to show you how this notation, which is what
I am going to do, off the side of these matrices helps you to find your errors.
So, I didn't want to have you watch me stare at the board, but right here and some of you
are probably screaming at me and making comments that you have made a mistake...stop...stop.
Row two plus three times row three, yeah when I came up here and I said let's multiply this
third row by three, zero times three is zero, zero times three is still going to be zero,
one times three is three, two times three is six, two times three is six, and then one
times three is equal to three.
You know when you are watching and you are not distracted, seeing these kind of errors
are easy.
But, you know, I am trying to talk through this as well.
But I still make these kind of mistakes in my work.
It is just frustrating, but it does happen.
You know, it happens to all of us.
Just be slow and meticulous with your work and give yourself someway of checking this.
Let's try this again.
Zero, two, that is still going to be zero, we are looking two and six is eight, six and
one is seven, and then three and zero is supposed to be three.
Now, that means that on this next step of work, I didn't make any notation in the second
row because I left it the same.
Zero to zero, zero to zero, eight, seven, and that is going to be three.
On this line of work, I worked with row and row two, so not only is this of course needs
to be changed to three, but that is going to change some possible answers here.
Now I just took row two and I didn't multiply anything and that is why there is no blue
marks up here.
We have negative two and zero is negative two.
Now we have zero and zero, and that is still going to be negative two, negative eight and
seven is still negative one, but negative four and three is going to be equal to negative
one.
All right, now we almost have the left side equal to an identity matrix, but of course...you
know..you are values of two and not values of one.
We are going to come up here and do one more step of work.
We are going to do negative one-half of row one and we are going to do one-half of row
two.
We are going to leave row three alone because that is zero, zero, one is what we need.
Our inverse matrix, we are going to need to check it to make sure that it is an inverse,
but we have negative two divided by negative two is one, zero, zero, negative two divided
by negative two is equal to one, and now we are looking at one-half, and another one-half.
We have, let's see here, one-half times row two or divide row two by two and we have zero,
one, zero, eight divided by two is four, 7/2, and 3/2.
Our last row is already good so we have zero, zero, one, two, two, and one.
That should be right here, as far as our notes were concerned, that matrix B...the new matrix
on the right hand side, we have moved our identity matrix from the left to the....
Let's try that again, from the right hand side to the left hand side, this is matrix
B or hopefully it checks out to be that inverse matrix.
Now what I need to do is step off and I am going to make a note that this is matrix A,
this is hopefully the inverse matrix, show the multiplication in both directions, and
verify that...you know...that we get that identity matrix when we doing A times inverse
matrix A and the inverse matrix A times matrix A making sure that this really is the inverse.
BAM!!!