CHAPTER 05.17 : SYSTEM OF EQUATIONS: Can we divide two matrices?
In this segment weíll talk about can we divide two matrices letís suppose somebody says hey [A] times [B] is equal to [C] so letís suppose [A] times [B] is defined by the number of columns in [A] is same as the number of rows of [B] and then [A] times [B] is defined so weíre tempted to say that hey oh you know [B] is [C] divided by [A] but we donít define matrix division like that when we are talking about matrices so what do we do is there something that is equal to dividing to matrices yes there is letís suppose if you have numbers, weíre talking about scalars now, if you have real numbers a divided by b a divided by b. We will see that a divided by b can also be written as a times b inverse so we can take number a divided by b and if we want to and if we want to find a times b it can be rewritten into a multiplication of two numbers where youíre multiplying a times b inverse sub b and of course there has some mechanism of finding this inverse sub b to be able to change this division into multiplication but thatís the point right here that hey there are similar effort here for matrices.
So in a matrix what we do is in matrices what we do is we say if a [B] is a n by n matrix so weíll say if [A] is a n by n matrix if [A] is a n by n matrix then [B] which is again an n by n matrix is the inverse of [A] so we are now talking about how we define the inverse of a matrix now you are already finding out that hey we are limiting our definition of the inverse of matrix by it is limited only to square matrices so if [A] is a n by n matrix a square matrix then [B] is the inverse of [A] if [A] times [B] is equal to the anti- matrix so thatís all we have to do so if [A] is a n by n matrix is a square matrix then you are able to find another matrix [B] we can call it the inverse of the matrix [A] if your [A] times [B] is called the anti-matrix now keep in mind that every square matrix the first thing which we will have to realize is that only for square matrixes does the inverse exist and the inverse does not exist for every square matrix weíll see that later that whenever somebody gives you a square matrix you wonít be able to every time find another square matrix by which when you multiply [A] by [B] that you get the anti-matrix.
So thatís the definition of inverse matrix so weíre also are able to say that hey if [A] times [B], [A] and [B] are both square matrixes then we will find hey that this is the identity matrix then [B] times [A] automatically will be also equal to the identity matrix. Many times people think that hey in order to show that one matrix is [A] is [B] is inverse of [A] we have to show that [A] times [B] is [I] and [B] times [A] is [I] no thatís not true if [A] times [B] turns out to be [I] then automatically [B] times [A] will be also equal to [I] so in order to show that one matrix inverse the other either you can multiply† the two matrixes in either fashion either [B] times [A] or [A] times [B] and if you get the anti-matrix then they are inverse of each other so this also makes it easier to understand that if [B] is the inverse of [A] then [A] has to be the inverse of [B] if [A] inverse exists we call [A] to be invertible or non-singular so if [A] inverse exists then [A] is called to be invertible or non-singular. If [A] inverse does not exist then [A] is considered to be singular, not invertible, not invertible or singular these are some words or some phrases which are used about inverse of matrices and things like that.
So the bottom line is that we are not able to divide two matrices itís not defined but we can always find the inverse of a matrix the first thing about finding the inverse of a matrix is it has to be square and then that inverse has to exist and how do we figure out what it means to find the inverse of a matrix is that the multiplication of the two matrices has to equal the anti-matrix that if [A] is the inverse of [B] or [B] is the inverse of [A] either this has to be true, or this has to be true, or if this is true, then weíll find this to be true, if this is true, then this will be true and thatís how you are able to establish whether one matrix is the inverse of the other. And thatís the end of this segment.