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’s suppose somebody gives you the following simultaneous linear equations. So when somebody is giving you three simultaneous linear equations and let’s 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’ll 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’ll say A, B, C. And then we’ll 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’s 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’m going to do is I’m going to put them in this kind of a matrix. I’m going to say hey I’m 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’m 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’ll 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’ll get this second row first column of this matrix here. The same thing I’ll 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’s 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.