Transcript FP2 MEI Lesson 5 Matrices part 2 solving simultaneous equations

the Further Mathematics network
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the Further Mathematics network
www.fmnetwork.org.uk
FP2 (MEI) Matrices (part 2)
Solving simultaneous equations
Let Maths take you Further…
Solving simultaneous equations using matrices
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
The determinant of a 3×3 matrix
The inverse of a 3×3 matrix
Example:
This system could be solved by
eliminating one variable, say z,
between two different
pairs of equations. But we’ll use
matrices.
Geometrical interpretation:
This will be the
point where
three planes
meet
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.
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.
Matrices and Simultaneous Equations
det M =0
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.
Elliminate z
These are identical lines, so the planes
form a sheaf.
To find a parametric form of the solution, let x = t (a parameter).
Writing the solution in terms of parameters
Independent study:
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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.