CHAPTER 04.01 - 04.05: PRIMER ON SIMULTANEOUS LINEAR EQUATIONS: Inverse of a Matrix
In this segment, we'll talk about the definition of an inverse of a matrix. I'm not going to show you how to find the inverse of a matrix, because that's in a separate segment, all we're going to do is to define what a . . . what an inverse of a matrix is.
So, first, the inverse of a matrix is only defined for a square matrix, so let A be a square matrix. So let A be a square matrix, so, which it is, because I'm telling . . . I'm telling you that, hey, it has n rows and n columns, same number of rows and same number of columns, so let A be a square matrix, then if there is a B, which is also n-by-n matrix, such that you find out that B times A, that you are . . . when you multiply B and you multiply it by A matrix, and it turns out to be the identity matrix, then B is considered to be inverse of A, B is inverse of A. So that's all you have to do is, in order to be able to figure out whether one matrix is inverse of another, and it's the definition of inverse of a matrix. So let's suppose somebody gives you a square matrix, and then says that, hey, there is a matrix B, that when you multiply the B matrix by the A matrix which was given to you, and it turns out to be an identity matrix, keep in mind again that the identity matrix also will be a square matrix, which it will have diagonal elements which will be 1 and the off-diagonal elements which will be 0, that's what an identity matrix is, and also will be the same size as n-by-n, because all of these, this n-by-n, this n-by-n, so the resulting multiplication matrix will be also n-by-n, then B is called . . . B is called the inverse of A. Now, keep in mind that this B matrix will not always exist for this A matrix. So, that's why we say that, then if there is a matrix. So always keep in mind that if somebody gives you a square matrix, it does not necessarily means that the inverse of the matrix exists, and there are certain ways which you can prove, or which you can identify, to show that whether a particular matrix has an inverse or not, and those are the kinds of this which we are not going into right now, but we wanted to be able to get through the definition of the inverse of a matrix. Now, what happens is that if you get B . . . if you get the n-by-n matrix B to be multiplied by the A matrix, and it turns out to be the identity matrix, and then you also find out that A times B will also turn out to be the identity matrix, and this is one place where the multiplication of two matrices turns out to be commutative, so that will also turn out to be the same identity matrix. What this means is that if B is the inverse of A, then A is the inverse of B, that's all there is to it. Many times, in some books people say B times A is equal to the identity matrix is equal to A times B in order to show that, hey, B is the inverse of A, that's not what you have to do, all you have to show is that B times A is the identity matrix, then this is automatically can be proven, that A times B will be the identity matrix. Now, many times when the matrix inverse exists, then this . . . these two matrices are called to be invertible, they may . . . they may call them invertible, that, hey, they can be inverted, that's where that word comes from, or people might say, hey, they are nonsingular. So, those are some of the other terms which are used for a matrix which can be . . . which . . . for which an inverse exists, that the matrix is invertible, or the matrix is nonsingular, and so on and so forth, but here what we want to be able to do is we want to be able to see that what does it mean to have the inverse of a matrix, that's all we are trying to show here.
So, for example, if somebody gives you . . . gives you a matrix, so if somebody says, hey, find if -5.5, 2.0, 1.5, -0.5, is inverse of 2, 8, 6, 22. So, keep in mind that in this segment I'm not showing how to find the inverse of a matrix, that's shown in a separate segment, what I am trying to give you as an example is that, hey, can you find out if this matrix is inverse of this matrix, so what that means is that when I multiply this matrix by this matrix, that if I don't get the . . . if I get the identity matrix, then it is . . . then it is the inverse of the matrix, otherwise it's not. So, all I have to do is go ahead and multiply these two matrices, so I get -5.5, 2.0, 1.5, -0.5, then I have 2, 8, 6, and 22. So if you follow the rules of matrix multiplication, this is what you're going to get. So that's your homework, matrix multiplication is shown in a separate segment, you get 1, 0, 0, 1. So you get this times this equal to the identity matrix. So that means that this is, in fact, the inverse of this matrix, and vice versa also, this is inverse of this matrix, automatically, so if you take this matrix and multiply it by this matrix, you're also going to get 1, 0, 0, 1, so that means that if I take this, 2, 8, 6, 22, and multiply it by the other matrix, which is -5.5, 2.0, -1.5 . . . 1.5, -0.5, you are going to get, again, the identity matrix, 1, 0, 0, 1, so they are the inverse of each other. Now let's go ahead and see that how does it help us to be able to figure . . . how does this inverse of a matrix help us in solving simultaneous linear equations? So let's suppose somebody gives you . . . gives you a simultaneous linear equation, and says that, hey, go ahead and . . . you have two simultaneous linear equations given to you, and you want to be able to find out what the value of the unknowns is, let's suppose 2, 8, 6, 22, and somebody says, hey, you've got two equations, two unknowns, x1 and x2 are the two unknowns, and your two equations are like this. How does the matrix inverse help us to figure out what these values of x1 and x2 will be? Now, going back to our . . . going back to our symbols, let's suppose, let's write it as a symbol, you've got A X equal to C, so A is your this matrix, X is this, the unknown, and C is the right-hand side. So, in this case, what you can do is you can take the inverse of . . . you can multiply both sides by the inverse of the matrix if that inverse exists. So, keep in mind that this multiplication which I am showing here is that, that this multiplication is allowed only if the inverse of A exists, so that's how I'm multiplying A inverse here and A inverse here. And, but, A inverse times A is what? Is the identity matrix. So identity matrix times X is equal to A inverse times C, so I get X is equal to A inverse times C, because when you multiply an identity matrix by any other vector, if the multiplication is allowed, then you get the . . . you get the matrix itself. So whenever you multiply an identity matrix by another matrix, and if the multiplication is allowed, then you get the second . . . second matrix itself. So what that means is that if I know the inverse of the A matrix, then I can just multiply it by my right-hand side, which is the C vector, which is this vector right here, then I can find out what the solution is, so that's what I'm going to do. So what that means is that x1, x2, is equal to 2, 8, 6, 22, the inverse of that, times 46, 128. So, that's how sometimes the inverse is shown, like A inverse, like this, that's how you will show the inverse of a matrix, A raised to the power -1. So, that's what I'm doing here, I'm just moving . . . moving this A matrix to the right-hand side, by taking . . . using the inverse of that. And now the inverse of this particular matrix, I already gave it to you in a previous . . . in a previous . . . in this particular segment I gave it to you, and that was -5.5, 2.0, 1.5, and -0.5, and I'm going to multiply it by the right-hand side, which is 46 and 128. And, once I do that, I get . . . when I multiply this matrix with this matrix, I get 3 and 5. Again, the multiplication of two matrices is covered in some other segment, and you can do this as a homework, to look at whether the multiplication turns out to be 3, comma, 5. So that's how you are able to . . . what the definition of the inverse of . . . the inverse of a matrix is, and how you are able to use it to solve a set of equations. So that's the end of this segment. |