 Once we have products of matrices, we can also talk about the inverse of a matrix. This requires introducing a few terms. Once we have matrix multiplication defined, we can define the identity matrix. Mathematicians like to recycle concepts. So our definition of identity comes right from the definition of field, where we had a multiplicative identity where 1 times a gave us a. Now since the multiplication operation in a field is commutative, that also means that a times 1 will be equal to a. And so that suggests the following definition for matrix identity. The identity matrix i is a matrix where a i equals i a equals a for all matrices a. Unfortunately, that's impossible. And that's because we can't always multiply two matrices. And even if we can multiply a times i, there's no guarantee that we can find i times a. So if our definition is to be about any matrix at all, we have to add a qualifier provided these products exist. And it's worth spending a few minutes thinking about the required properties of such an identity matrix. I'll wait. After thinking about this, we come to the following theorem. Let a be an n by n matrix, where the entries of a are 0 if i is not equal to j, and 1 if i is equal to j. Then this matrix is an identity matrix. And you should be able to prove this. And as an example, a 3 by 3 identity matrix is a matrix whose entries are 1, if the row and column numbers are equal, which puts them along the main diagonal, and 0 everywhere else. So suppose I have some matrix m. The left inverse of m is a matrix where a m is equal to i, the identity matrix of the appropriate size. And the reason that this is called the left inverse is because a is being multiplied on the left of m. So of course the question you're asking is could there be a right inverse of m? And the answer to that is, sure, why not? The right inverse of m is a matrix B where m times B is going to give us i, the identity matrix of the appropriate size. And so here we're multiplying by B on the right and we're getting a right inverse. Well what if I have a matrix C where if I multiply on the left, I get Cm, or if I multiply on the right, mc. In either case I get the identity matrix. In that case, I'll call C the inverse of m, no right, no left, just plain inverse. And we frequently indicate the inverse of the matrix m as m superscript minus 1 by analogy with our multiplicative inverse. But it's important to recognize that there is in this case no indication of the quotient 1 divided by m. That is meaningless in the context of matrices. This is just the notation because there are only so many symbols that says that we're looking at the multiplicative inverse of the matrix m. So let's introduce a couple of terms here. If a matrix has an inverse, it's called a non-singular matrix. Otherwise it's a singular matrix. And these automatically include any non-square matrix. They also include any square matrix that doesn't have an inverse. So how do you find the inverse of a matrix? We can find the inverse of a matrix by solving a system of equations. Since it's enough to find either a left or a right inverse, we'll let the entries of our inverse matrix X be variables. And the equation that we'll have is m times X should give us the identity matrix. Let's see how that might work. So I have a 2 by 2 matrix, and let's see if I can find the inverse of this matrix. So I want to multiply this by some matrix to get the identity matrix of the appropriate size. And since this is a 2 by 2 matrix, I know that if I multiply it on the right by any matrix, the product is going to have two rows. And because this should be the identity matrix and the identity matrix is always square, I know that the product will give me the 2 by 2 identity matrix. And now I can ask the question, what sort of matrix would I have to multiply by to get this 2 by 2 identity matrix? And since the number of columns in the product is the same as the number of columns in the second factor, that means the second factor must also be a 2 by 2 matrix. And so I have four unknowns, X1, X2, X3, and X4, and my matrix multiplication is going to give me a system of four equations. And I can perform row reduction on this system of equations. And after all the dust settles, and there is quite a bit of dust, but after all the dust settles, we get our reduced row echelon form of the matrix, and we can read off our solutions. X1 equals negative 3, X2 equals 5, X3 equals 2, and X4 equals negative 3. And so here we have the four components of our inverse matrix. Well, that was fun. Let's do another one. So let's try to find the inverse if possible. And the same sort of analysis we must be dealing with a 2 by 2 matrix, and our matrix multiplication gives us a system of equations, and we can find our augmented coefficient matrix, and we can reduce the augmented coefficient matrix. And we have a problem because if we look at this third row, it corresponds to the equation 0X1 plus 0X2 plus 0X3 plus 0X4 equals negative 2, but that's impossible. Since our system of equations has no solution, no inverse exists.