CHAPTER 05.02: SYSTEM OF
EQUATIONS: Writing simultaneous linear equations in matrix form In
this segment well talk about how we will write simultaneous linear equations
in matrix form. So let痴 suppose somebody gives you the following
simultaneous linear equations. So when somebody is giving you three
simultaneous linear equations and let痴 suppose they say - hey write this in
the matrix form, we seem to automatically jump to this. So if I asked
students to do this they will simply start writing like this. They値l say 25,
5, 1, 64, 8, 1. You got to understand the
coefficient of C here is one not zero. Just because
there is nothing there 144, 12, 1. Then we値l say A, B, C. And then we値l put
106.8, 177.2 and 279.2. But
the thing which we have to realize is that how do we get from here to here.
That痴 our direct step went from here to here. It might come through as a
direct step once you have a lot of experience with it but there is a lot of
things which go beyond before you get to this level. So one of those steps is
that what I can do is I have these three simultaneous linear equations so
each one of them is a separate equation. And what I知 going to do is I知
going to put them in this kind of a matrix. I知 going to say hey I知 going to
write 25a plus five b plus c right here and I can put 106.8 here. And I guess
here 64a plus eight b plus c here and I get 177.2 here. And I get 144a here
plus 12b here plus c here and I get 279.2. So what I have basically done is
that I have put these three simultaneous linear equations in some sort of a
matrix form. Where I got three rows in one column and three rows in one
column right here. And
I can rewrite this now, I can say hey what I can do is now I have it written
like this but if I write 25, 5 and one so I知 trying to separate out the
unknowns here. If I write 25, 5, and 1 here, I could write this as A, B and
C. And if I write this a, b, and c. And if I write this A, B and C you can
very well see that if I have 25, 5, and one and multiply it by this vector a,
b, and c I値l get the first row first column this part right here. Same thing
here, 64, 8, and 1. When I multiply 64, 8 and one by a, b, c by this row
vector by this column vector I値l get this second row first column of this
matrix here. The same thing I値l do 144, 12, and one. If I multiply this row
vector by this column vector I will get this third row first column right
here. And you can see that this is the three row three columns, and this is
three rows one column. And we multiply three rows and three columns by a
vector which is three rows and one column. I get three rows and one column
right here will be equal to 106.8, 177.2 and 279.2. So
it痴 important to understand that hey we go from this to here and then from
here to here through matrix multiplication. This is equal to this is based on
when do we consider two matrices to be equal when element to element they are
the same. This element right here is the same as this element as given by the
first equation, this element here is the same as this element here as given
by this equation. And this element here is the same as this element as given
by this equation. You also need to recognize that hey these equations can be
or these equations right here can also be written as a linear combination.
Like I can put a here, and I can put 25, 64 and 144 her plus b times 5, 8 and
12 here plus c times one one one
here and that will be equal to 106.8, 177.2 and 279.2. So we can look at
these three equations three unknowns also as that hey these sets of equations
can be used as a linear combination of this vector, this vector and this
vector of the three column vectors equal to this column vector. And whatever
linear combination of A, or values of A, B and C which would satisfy this
linear combination to be equal to this value this vector right here will be
the values of A, B and C. So you got to keep an open mind in terms of
recognizing what it means to convert a set of simultaneous linear equations
into a matrix form. And that is the end of this segment. |