Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 We will cover 3topics today 1. Homogenous Equations 2. Jacobi Iteration 3. Exam Questions

2. Lecture Recap We looked at the Gauss-Jordan method of inverting the matrix ‘A’. This uses an augmented matrix, e.g. Last lecture we looked at methods of solving linear equations. We took the equation And deduced that if we can find the inverse of A we can solve for the unknowns. I.e. Matrix multiplication is a row on column operation. Hence, by using row operations to turn the l.h.s into the identity matrix we can find the inverse of A.

3. Lecture Recap If we are only concerned with finding ‘x’ in I.e. if we start with Then we can use the row reduction method. Hence we do not need to compute the inverse. If the augmented matrix is And convert this to Then converting this matrix to Then we have found A-1 Gives us ‘x’

4. Homogenous Sets of Equations Consider the following set of linear equations R2 = R2 – R1 R3 = R3 – R1 x + y + z = 0 x + 2y = 0 x - 3y + az = 0 R3 = R3 + 4.R2 Use the row reduction technique to find the value of ‘a’ that gives an infinite number of non-trivial solutions. Thus a = 5 gives infinitely many solutions. Notice that Thus the determinant = 0 if a = 5

5. Homogenous Sets of Equations Any set of equations Ax = 0 is known as a homogenous set. It is a set of equations with zero right hand sides. Then, using Cramer’s rule Clearly, the equations have the trivial solution x = 0. There may also exist non-trivial solutions. Hence, if det A ≠ 0 then only a trivial solution exists. Recall that, if However, if det A = 0 then we can prove that an infinite number of non-trivial solutions exist.

6. Jacobi Iteration So far we have met a number of techniques for solving systems of linear equations. We will now examine a technique that provides approximate solutions to linear systems. We have used determinants (Cramer’s rule). We will be looking at the Jacobi method of iteration. We have the Gauss-Jordan inverse matrix method. With repeated application of Jacobi’s method the approximate solutions approach the exact answers. Finally, we have used the Row Reduction technique (Gaussian elimination).

7. Jacobi Iteration It is most convenient to explain the process using an example. Step 2 Rewrite the equations in the form Say we had the simple simultaneous equations Step 1 Where ‘n’ is the nth iteration Rearrange these to make x and y the subjects. I.e. Step 3 Make a reasonable guess at the answer. I.e. Notice that the coefficients of the variables are less than one.

8. Jacobi Iteration Step 5 Use these answers to perform the next iteration Step 4 Perform the first iteration Therefore thus After 12 iterations we get the exact answer (to 4 d.p.)

9. Exam Questions If the Jacobi iterative method is applied to the system of equations Starting from the initial guess x = y = z = 1, the result after one step is a) b) c) d) The rearranged equations are

10. Jacobi Iteration Question Find an approximate solution to Perform 5 iterations and take x0 = 0 and y0 = 3 as your initial guess The equations are rearranged to make x and y the subjects.

11. Jacobi Iteration

12. Exam questions For each of the following systems of equations (1) Express the system in matrix form (2) Construct the augmented matrix (3) Use the method of ROW REDUCTION to obtain the set of solutions, where it exists. x + 2y − z = 2 (b) x + y + 4z = 3 x + y + z = 3 2x + y + 2z = 4 3x + 2y + z = 1 3x + y = 5.

13. Exam questions a) (1) In the matrix form A.x = d we have (2) The augmented matrix is

14. Exam questions (3) Using the row reduction method with

15. Exam questions b) (1) In the matrix form A.x = d we have (2) The augmented matrix is

16. Exam questions (3) Using the row reduction method with Thus One of the equations is redundant but consistent. Hence, there will be an infinite number of solutions.

17. Exam questions Show that the system of homogenous equations x - y − z = 0 3x + y = 0 6x - 2y - 3z = 0 has infinitely many solutions, and determine the general solution in terms of a parameter t. Write the system of equations. x + 5y - 3z = 17 2x + 9y - 14z = 23 x - 5y + 5z = -5 in matrix form. Construct the augmented matrix and use the row reduction method to obtain the solution.

18. Exam questions If 3x + y = 0, then let x = t hence y = -3t. Finally z = 4t. The augmented matrix is Therefore; x = 5, y = 3, z = 1

19. Conclusion Essential reading Mathematics for engineers: Chapter 13, Block 5 Finish your HELM reading!!! We have covered 3topics today 1. Homogenous Equations 2. Jacobi Iteration 3. Exam Questions