1. the Further Mathematics network www.fmnetwork.org.uk

2. the Further Mathematics network www.fmnetwork.org.uk FP2 (MEI) Matrices (part 2) Solving simultaneous equations Let Maths take you Further…

3. Before you start: You need to have covered the work on Matrices in FP1, particularly the work on using matrices to solve linear simultaneous equations. You also need to be able to find the determinant and the inverse of a 2x2 matrix and 3x3 matrix. When you have finished… You should: Be able to solve a matrix equation or the equivalent simultaneous equations, and to interpret the solution geometrically Solving simultaneous equations using matrices

4. The determinant of a 3×3 matrix

5. The inverse of a 3×3 matrix

6. Example: This system could be solved by eliminating one variable, say z, between two different pairs of equations. But we’ll use matrices.

10. Geometrical interpretation: This will be the point where three planes meet

11. Singular matrices: det M =0 As in the 2d case, if this happens there is either (a)no solution, or (b) infinitely many solutions: Consider (a) No solution/equations inconsistent (i)No two planes are parallel. In this case the planes form a triangular prism. (ii) Two planes are parallel and distinct, and are crossed by the third plane. (iii) Two planes are coincident and the third plane is parallel but distinct. (iv) All three planes are parallel and distinct.

12. Singular matrices: det M =0 Now consider (b) Infinitely many solutions / equations consistent (i)No two planes are parallel. In this case the planes have a line of common points. This arrangement is known as a sheaf and the solution can be given in terms of a parameter. (ii) All three planes are coincident.

13. Matrices and Simultaneous Equations

14. We cannot find the inverse matrix so we need to use algebra. None of the planes are parallel, so there are two possible cases, the triangular prism and the sheaf. det M =0 Elliminate z

15. These are identical lines, so the planes form a sheaf. To find a parametric form of the solution, let x = t (a parameter).

17. Writing the solution in terms of parameters

19. Independent study: Using the MEI online resources complete the study plans for the section: Matrices 3 Do the online multiple choice tests for this section and submit your answers online.