In this segment, we are going to take an example and solve it by using Naive Gauss elimination
method.
So here you are given three equations; you're given three equations, three unknowns.
The three unknowns are a sub 1, a sub 2, a sub 3.
So we want to be able to find out what these three unknowns are.
Now, the first step - what you're going to do is we're going to take the three equations,
three unknowns which are given in the matrix form.
We're going to write the Augmented matrix.
This is called the augmented matrix.
This is going to make the explanation a little bit easier, and the difference between the
matrix form of the equations and the Augmented matrix is that you are writing down your coefficient
matrix here and then you're writing the right-hand side vector right here.
So you're not showing what the knowns are, but it is presumed that you understand there
are four, three rows, and four columns and the fourth column is your right-hand side
vector.
Now we have two steps in naive gaussian elimination.
One is the forward elimination step and the second is the back substitution step.
So let's go ahead and see that - how we can do these two steps.
So let's concentrate on the forward elimination.
So how many steps of forward elimination do we have to conduct?
It's basically n minus 1, where n is the number of equations you have, or n is the number
of rows and columns you have in the coefficient matrix.
In this case we have three equations, you subtract one from here to get two.
So it's important to understand the number of steps of formation is equal to n minus
1.
In this case, it is equal to a 3 by 3 matrix for the coefficient matrix, or three equations
three unknowns.
We'll have two steps of forward elimination.
So we're going to concentrate on step one of formation.
Our step one corresponds to is - that you're going to take the first column and go below
the first column, which is second row and third row, and you want to make these two
elements to be 0.
And how we're going to make these two elements to be zero is by using the first column - first
row - and taking multiples of the first row and subtracting it from the second row to
be able to make this 64 to be 0 and this 144 to be 0.
So, in order to make the second row, first column to be 0, I'm going to take a multiple
of 64 divided by 25 which would be equal to 2.56.
So I'm going to take the first row, multiply it by 2.56 and the element which I'm going
to get here is 64; and that means that when I subtract this whole thing, which I get as
a resulting multiple of the first row, I'm going to get zero here.
So here is the second row.
I subtract the multiple of the first row; 64 minus 64 gives me zero, because that's
what I was looking for.
And then the resulting row is given here.
What I want to do is I want to replace my second row by what I just obtained.
And here's my zero.
Now this is not the end of the first step of forward elimination elimination because
I have still have not made this to be zero.
So let's go ahead and see how I can make that to be zero.
Now I want to make that zero again.
I have 144 here, I need to make that zero; and this is my 25, so I'm going to make the
multiple to be 144 over 25, which is 5.76.
So I'm going to take the first row, multiply by 5.76, and here I get 144 because that's
going to give me the zero which I was looking for.
So that's the multiple of the first row.
So this is my third row, I take the multiple of the first row which I just found out and
I subtract the two.
And this gives me zero, which I was looking for.
And this will be the resulting row which I obtain.
I'm going to replace my third row by whatever I get there.
I have ended the first step of forward elimination because I've gotten zeroes below the - below
the first row in the first column.
What is left now is the second step because I have only two steps of forward elimination
to conduct.
And what I need to do is I need to look at the second row and make everything below the
second row in the second column to be 0.
In this case it turns out to be - I just have to make that to be 0.
So the multiple will be -16.8 divided by -4.8, because i want to take this number which is
3.5.
So take the second row which I have multiplied by 3.5 and I will get this as my resulting
answer there.
I'm going to write down my third row; we will subtract the multiple of the second row and
because this number is same as this number and I subtract the two, I'm going to get 0.
And that's what I was looking for.
So the resulting third row which I obtained is going to be replacing my third row in the
Augmented matrix.
And that's the end of the steps of all forward elimination because I have been able to reduce
the coefficient matrix, which is right here, the coefficient matrix is right here but I
am able to reduce it to an upper triangular matrix.
You going to get a 0 here, 0 here, and a 0 there.
So that means that's the end of the forward elimination steps.
So let's go ahead and see now how we can do back substitution to be able to find out what
the unknowns are.
So this was the Augmented matrix which I obtained at the end of the forward elimination steps.
I'm going to rewrite them back into the matrix form so that I can show you how to do the
back substitution.
If you look at the last equation right there, it can be written as 0.7 a3 is equal to 0.7.
So you only have one unknown in the last equation, and that gives you a3 is equal to 1.
Now let's look at the second equation.
The second equation is obtained from these elements and this element is the right-hand
side.
So you have -4.8 a2 minus 1.56 a3 is equal to -96.2.
However, we don't have two unknowns in this equation here, we only have one unknown because
we just found out a sub 3.
So we're going to take this to the right-hand side and this is what I'm going to get by
writing down the equation in terms of
a
sub 2.
I substitute