Hi, everyone. This is Professor Yong-Jin Yoon from KAIST. This is third session of week 5 for the basic mathematics for the beginner of AI, linear algebra. Up to last session, we studied about how to calculate determinant of A by using the row operation to theory. From this session, we are going to study about how to calculate determinant of A by using row operation with example. Here, I bring out the 5 times 5, the square matrix, which is very big. Then very [inaudible] how to calculate this, the determinant of A by using just a rule of defining the determinant A. However, once you change this A to the upper triangular matrix, we can find the determinant of A very easily. This is all the row operation I used to reduce A to the U. Those are how many step? Nine steps to change matrix A to the matrix U here. Then to find the determinant of A, we changed the interchange rows in the first, so we put the minus one here. Also second process, we change a row and then our four becomes R_4 minus R_1. We multiply R_5 is what, one here. R_3 becomes R_3 plus R1. R_5 is one here too. R_4 becomes 2 times R_4 plus R_3. We multiply two here. Then R_5 become 2 times R_5 minus 3 R_2. We multiply 2 here. Then R_4 becomes R_4 minus 5 R_3. We just are R_5's here, we just multiply one here, one here, one here like that. Then times determinant A becomes determinant of U. Determinant of U is what? They're just product of the diagonal element of U minus 1 times 2 times 1 times 1 times minus 8. You can calculate determinant of A by using this row operation. So far, we have been studying about how to calculate determinant of square matrix by using the row operation. We are going to move on to the eigenvalue problem. We have big picture. When you solve the system of linear algebra equation, AX equal P, we use tablet form A bar B. Also, when you find the inverse of matrix A, we also tabular form A bar I. In the last way, when you find the determinant of A, we also use row operations. All the process, we use row operation. Then A bar B becomes U bar C, upper triangular matrix. Finding inverse of matrix also can use it. When you find the determined to A, you also use row operation. By using the determinant of U is very easy to calculate. We can find determined A from the determinant of U. If all the diagonal element of you are not zero, then AX equals B has a unique solution and it means A is invertible and determinant of a is not zero. Because if there is any zero in the determinant of U, zero on the diagonal element, then it become zero. If the element of U is not zero, diagonal element of U are not zero, then determinant A is not zero. It means AX equals B has a unique solution and A is invertible. If upper triangular matrix U has a zero diagonal element, then determinant of A is zero and A does not have an inverse matrix, and AX equals B has either no solution or infinitely many solutions. Those three are connected to each other. As I introduced in the first part of our class, solving AX equals B and finding inverse matrix of A, and finding the determinant of A, all are related to each other. If the determinant of A is 0, then there is no inverse matrix of A and then there's maybe no solution or many solution. Those are the relationship between those three area. Here we can say something about determinant of A and the solution of the homogeneous system equation AX equals 0. If determinant of A is not zero, then AX equals 0 has a unique solution, which is X equals 0. If determinant of A is zero, then AX equals 0 has infinite many solutions. If AX equals 0 has a unique solution, then X equals 0, and it means determinant of A is not zero because it's unique solution. If AX equals 0 has infinitely many solution, then determinant of A is zero. This is very important when you study about the eigenvalue problem. Please remember that if determinant of A is zero then AX equals 0 has infinitely many solutions. I bring out some example for the linear independence problem. Are those vectors, four vectors are linearly independent. It means towards the below, the homogeneous system of equation has a unique solution. But this is related to each other. When you form this, C_1W_1 plus C_2W_2, C_3W_3 plus C_4W_4 equals 0. If the system has a unique solution, then the vectors are linearly independent. Unique solution means that C_1, C_2, C_3, C_4 equals 0. Otherwise, they are linearly dependent. We can formulate this 4 by 4 like a matrix form for the homogeneous system of linear algebra equation. C_1 plus C_2 plus C_3 plus C_4 equals 0, and 2C_1 plus 2C_4 equals 0. 2C_1 plus 3C_3 plus 7C_4 equal 0, and C_2 minus C_4 equals 0. Here, if we want to know that those four vectors are linearly independent, we need to find that whether this system of linear algebraic equation has a unique solution or not. If we have a unique solution, then those four vectors are linearly independent. Unique solution means that, C_1, C_2, C_3, C_4 equals 0. How to check whether this system of linear algebraic equation has a unique solution? There is a way we can calculate the determinant of A, and then if determinant of A is not zero, then this one has a unique solution. Let's calculate the determinant of this matrix A. I just choose here, luckily there is a third column which has a lot of zero here. I choose third column here and then I can easily calculate determinant by using the definition of determinant and minus 1, 2, 3 plus 1 times 1, and determinant of the 2,0, 2 2, 3, 7 0, 1 minus 1. Then I can easily calculate that this is minus 16. Determinant of A is not zero. It means that this one has a unique solution, which is C_1, C_2, C_3 equals 0. This system has a unique solution, so the given vectors are linearly independent. Up to here, we study about the relationship between the determinant of matrix is whether it is zero or not, and it is related to whether there is a unique solution or the infinite many solution, and no solution. Also inverse of matrix, invertible or not. Then I end up with a homogeneous system of equation. Thank you very much and see you in the next session.
