Hi everyone. This is Professor Yongjin Yoon from KAIST. This is second session of the Week 4, or the basic mathematics for the beginner of AI, Part 1, Linear Algebra. Let me review again for the big picture of our course. We start with the review of matrix, and we already finished to define the solution for the system of linear algebraic equation. Also, just we finished the part and then during the time, we start our linearly independent vectors. Now, we can move on to the next session, inverse of square matrix A by using the row operation. After that, we're going to study about determinant of square matrix A. From there you can study about a matrix Eigenvalue problem and diagonalization problem and AI application for deep learnings and support vector machine application. Let's start second part of our major content, the inverse of square matrix. What is the definition of square matrix? Let's say if you have B, N by N square matrix B is said to be an inverse of matrix A, A also be N by N matrix. If the relation AB equals BA becomes identity matrix I. This I is N by N matrix too. If you can find the matrix B, which satisfy AB equals BA equals I, then, this B, we call that is a inverse of matrix A. Here, for example, 4 by 4 identity matrix is become like this, so how many inverses can square matrix have? Only one, and most one. It is possible that a square matrix may not have an inverse matrix. For example, 1, 2 2, 4 does not have an inverse matrix. How to prove that? Let's say inverse matrix of 1, 2 2, 4 is a, b, c, d. From the definition of inverse matrix, if we multiply that inverse matrix to the matrix A, here matrix A is 1, 2 2, 4, then it should become the identity matrix. Here, 1, 2 2, 4 times a, b c, d equals what? 1a plus 2c,b plus 2d and 2 times a plus 2c, and 2 times b plus 2d like that. To satisfy the definition of inverse matrix, 8 plus 2c should be the one, and then the element in the second row in the first column, is 2 times a plus 2c is 2. This 2 times a plus 2c should become 0, to satisfy the definition of the inverse matrix. In this case, 1, 2 2, 4 doesn't have the inverse matrix. If a square matrix A has an inverse matrix, then there is only one inverse it can have. How to prove it? Let me prove this one. Let's say B is an inverse of a matrix A. Then from the definition, AB equals BA equals identity matrix from the definition of inverse matrix. Let the matrix C be an inverse of matrix A. Then from the definition we also can say AC equals CA equals identity matrix. From the AB equals BA, we can multiply matrix C to the right side of AB, with AB and BA, then we come AB times C equals BA times C. If we look at A times B, then it becomes what? I. I times C left-hand side become I times C. If you look at in the right-hand side, BAC only look at the AC, then AC is what? Because C is our inverse matrix of A, it becomes I. In the end it becomes C equals B. We set up the two different inverse matrix B and C, but in the end, we found that B and C is equal. There is only one inverse. If the square matrix A has an inverse matrix, then we prove that there is only one inverse matrix.The other one is if A has an inverse, we say A is invertible. In that case, the inverse of A is denoted by A to the minus one. So A times A to the minus 1 equals A to the minus 1, A equals I, or that we say A times A inverse equals A inverse A times A equals identity matrix. If A is invertible, how can you find A inverse matrix of A, and if A is not invertible, how can you know that? We can form the tabular form A bar I and perform the row operation and we can see that whether the A has inverse matrix or not. We can formulate tabular form A bar I and then change it to the I var V. Then we try to change the original matrix A to the identity matrix I, and then with this row operation, the right-hand side Identity matrix I, change it to the V from certain matrix. If this row operation is possible to change, then A is invertible. If A can be reduced to the I through the row operation, then V is the inverse of A. If A cannot be reduced to the identity matrix, then A is not invertible. The fold of proof is very not easy, so for the highly motivated students, please try to do that, and then I introduce here for the proof for this, whether we can check the inverse of a matrix, it can be obtained from the checking of this row operation. Up to here, we study about what is definition of inverse of matrix and how we can find the inverse of matrix A by using the row operation. For the next week, our next session, we are going to study with example. Thank you very much.
