Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 We will cover 4topics today 1. The Inverse Matrix 2. Gauss-Jordan Inversion 3. Gauss-Jordan Elimination 4. Linear Systems of Equations

2. Lecture Recap The product of a row and column vector is calculated as Last week we learnt how to multiply matrices. Remember that not all matrices can be multiplied, the two matrices must have the right shape. They must be conformable. If A is an m x n matrix and B is a p x q matrix then A.B will be a m x q matrix provided that n = p If we wish to calculate the product of a larger matrix. Each element is calculated following the procedure above. However, we must use the appropriate row and column of the matrices A and B respectively. (m x n) (p x q) = m x q

3. The Inverse Matrix If a system of linear equations are represented by Suppose that A and B are both square matrices of the same order that satisfy Where ‘A’ is a matrix of coefficients, ‘x’ and ‘d’ are vectors. If there is a matrix ‘B‘ that exists with this property then B is the inverse of A. B is uniquely determined by A. Therefore we can write that Let Then NOT

4. The Inverse Matrix Thus If and Hence, Then We can easily deduce that

5. The Inverse Matrix To form the inverse the elements on the leading diagonal are interchanged. The remaining elements are multiplied by -1. Recall that if then If If we compare this with Then the matrix has no inverse and it is said to be singular. However, if Then clearly the inverse equals Then the matrix has an inverse and it is said to be non-singular.

6. The Inverse Matrix Question If Find A-1, B-1, B-1.A-1, (A.B)-1

7. The Inverse Matrix Properties The inverse of the identity matrix is the identity itself If we take the inverse twice we get our original matrix The determinant of the inverse is equal to the inverse of the determinant The inverse of the transpose is the transpose of the inverse

8. The Inverse Matrix The inverse of the product of two matrices is equal to the product of their inverses in reverse order (assuming the products exist) If an invertible matrixis symmetric or anti-symmetric, upper triangular or lower triangular, or diagonal then its inverseis of the same form. Square matrices that have the following property are called orthogonal matrices and have determinant equal to ±1

9. The Inverse Matrix So far we have found the inverse of a second order square matrix. This is a simple case and the calculation of the inverse becomes more complicated as the size of the matrix increases. Recall that if then If A is a 3rd order square matrix (therefore x and d are column matrices with 3 elements) then we can prove that This calculation would obviously take a long time. Hence, we want a method that calculates this inverse more efficiently.

10. The Inverse Matrix If Find A-1 Question

11. Gauss-Jordan Inversion Consider the following equality Now change ‘A’ by adding the 2nd row to the first row. This gives Matrix multiplication is a row on column operation. Hence, any row operation applied simultaneously to both ‘A’ and ‘I’ maintains the equilibrium. I.e. if What is A2.B? This is the result obtained by taking the second row of the identity and adding it to the first row.

12. Gauss-Jordan Inversion Thus, if This is called the augmented matrix And (obviously) Then we can use row operations to transform ‘A’ into ‘I’ on the l.h.s. By simultaneously performing the same row operations on ‘I’ (on the r.h.s.) we obtain A-1. Hence, if We can find A-1 in the following manner.

13. Gauss-Jordan Inversion Find the inverse of

14. Gauss-Jordan Inversion

15. Gauss-Jordan Elimination Earlier we took the the following equality We have learnt how to find the inverse of a square matrix. This helps us solve a system of linear equations. I.e. if we have And deduced that any row operation applied simultaneously to both ‘A’ and ‘I’ maintains the equilibrium. This same logic can be used with We can now solve this equation and find x in the following way. I.e. we can use row operations to change this equation into However, this is still a long winded way of solving for x. How can we solve for x if we don’t want to compute the inverse. Which directly gives us x. This is sometimes called the Row Reduction method.

16. Gauss-Jordan Elimination Example: Find x, y & z If In matrix form this is expressed as The augmented matrix is

17. Gauss-Jordan Elimination Thus, x = -1; y = -2 & z = 2

18. Gauss-Jordan Elimination Question Solve In matrix form this is expressed as The augmented matrix is Thus, x = 2; y = 3 & z = -1

19. Linear Systems of Equations ii) An infinite number of solutions When a system of linear equations is solved, there are three possible outcomes. Either Let y = λ then x = 1 – 2.λ i) A unique solution In this case ‘λ’ is called a free variable. ii) An infinite number of solutions iii) No solution. iii) No solution i) A unique solution A trivial example is We say that these equations are incompatible. This has the solution x = 1 & y = 2

20. Linear Systems of Equations iii) No solution The previous example was very easy. Sometimes it is more difficult to detect whether or not unique solutions exist. Consider The augmented matrix is Row 3 can not be satisfied hence these equations have no solution. We say that row 3 is inconsistent. We can tell that there is no unique solution because det A = 0

21. Linear Systems of Equations If the determinant of a linear system of equations is zero then we can not find a unique solution. However, sometimes we can find an infinite number of solutions. Question Solve: Row 3 is now consistent. Hence,

22. Conclusion Essential reading for next week HELM Workbook 7.4 Inverse of a Matrix HELM Workbook 8.2 Solution by Inverse Matrix HELM Workbook 8.3 Solution by Gauss Elimination We have covered 4topics today 1. The Inverse Matrix 2. Gauss-Jordan Inversion 3. Gauss-Jordan Elimination 4. Linear Systems of Equations