Module title & Course code
Matrices and Geometry
MAS 171
Lecturer
K. Ohktiani
Course description
University course content
Solving simultaneous equations by Gaussian
elimination
This unit will introduce the concept of a matrix, along with basic rules for arithmetic with matrices; it
will also demonstrate the use of matrices for systematic manipulation of linear equations. It will
introduce the important notions of determinants, eigenvalues and eigenvectors, and some particular
classes of matrices with special properties. The unit will also introduce basic ideas of co-ordinate
geometry, particularly Cartesian co-ordinates, emphasising links between geometry and algebra. It will
cover formal mathematical representations of straight lines and of circles and other conics, and
techniques for obtaining related geometrical objects such as tangents.
Further
A Level course content
Module
module
only?
Although Gaussian elimination itself does not feature on the A level
FP1/FP3/D2 Y
specification, students of modules FP1 or FP3 will be familiar with the idea of
turning a system of simultaneous equations into a single matrix equation.
Students of module D2 are perhaps best prepared for this as the augmented
matrix will come as little surprise to students who have seen a simplex tableau.
The simplex method also involves using row operations in exactly the same
way. Students of FP3 may have some general appreciation of the usefulness of
triangular matrices as they may have calculated their determinants.
Linear dependence and Independence
These are completely new concepts to all students. Students of FP1/FP3 may
have and advantage as they should know the abstract definition of a linear
transformation so will be familiar with the word “linear” being applied to a
linear combination of vectors. Few students, however, will have a deep or
secure understanding of this area of the specification as it is rarely required.
All students of C4, however, should have little trouble adding vectors and
multiplying by a scalar.
FP1/FP3/C4
Y
Reduction to Row Echelon Form
Students of module D2 will be used to reducing matrices to a different form via
row operations. Row echelon form itself will be completely new to all students
D2
Y
Algebra of matrices
Students of modules FP1 and FP3 should be able to add, scalar multiply and
matrix multiply matrices of any size. They should know about the dimensions
of a matrix. Subscript notation is not used to describe the entries of a matrix.
Students are unlikely to know the words “associative” and “distributive” but
may well know the word “non-commutative” and should at least know that
matrix multiplication is not commutative. All students study module C4 and
will understand vectors - but they are always column vectors as defined here
(row vectors as defined here would only be used to denote the co-ordinates of
a point at A level). As mentioned earlier, students of FP1/FP3 should be
familiar with the matrix form of a system of simultaneous linear equations.
FP1/FP3/C4
Y
Transpose of a matrix
Only students of module FP3 will know how to take the transpose of a matrix.
They should also know the definition of “symmetric” but not “skewsymmetric”. Although they should know that (AB)T=BTAT, they will not know
that ATA is always symmetric (proofs of this type are not beyond the scope of
FP3, though).
FP3
Y
Identity and inverse matrices
The identity matrix will be familiar to students of FP1 and FP3. Students of FP1
can form 2x2 inverse matrices. Students of FP3 know the word “diagonal” and
can form 3x3 inverse matrices. Students of FP1 know the words “singular” and
“non-singular” and how these are connected to the determinant.
FP1/FP3
Y
Gauss-Jordan elimination
It would be rare but not impossible for a student of module FP3 to know this
method. Again, students of module D2 have an advantage in that they can do
the simplex algorithm. In the 3x3 case, students of module FP3 are likely to
know how to form the matrix of cofactors and the adjoint matrix so the
formula given should not be a surprise.
FP3/D2
Y
Determinants of matrices
Students of FP1 can calculate a 2x2 determinant. Students of FP3 can calculate
minors, cofactors and 3x3 determinants. Neither students are likely to have
considered these as denominators arising from solving simultaneous
equations. Students of FP3 are extremely unlikely to know the effect of row
FP1/FP3
Y
operations on the determinant. However, it is likely that students of FP3 will
know how easy it is to calculate the determinants of upper/lower triangular
and diagonal matrices.
The vector cross product
Students of module FP3 should be able to calculate the cross product and
should know the formula given in terms of 3x3 detereminants
FP3
Y
Application of determinants to linear
independence
This will be new to all students
N/A
N/A
Forming the adjoint matrix
All students of module FP3 know this method for 3x3 inverse matrices
FP3
Y
Eigenvalues and Eigenvectors
Students of FP3 know the Eigenvalue and Eigenvector equation. They may use
the words “characteristic polynomial” to refer to the Eigenvalue equation.
They will not know the trace of a matrix
FP3
Y
Linear independence of eigenvectors
This is new to all students. However, students of module FP3 do know the
related fact that the eigenvectors of symmetric matrices are orthogonal. They
will not have considered eigenvectors of iterated matrices.
N/A
N/A
Symmetric and Orthogonal matrices
Students of FP3 should know that the eigenvectors of a symmetric matrix are
orthogonal. They will not have considered the possibility of non-real
eigenvectors. They should know the meaning of “orthogonal” and
“orthonormal” when applied to vectors. They should know that an
“orthogonal” matrix has orthonormal columns (and as such be clear on the
different usage of the word “orthogonal for matrices and vectors). They
should also know that the transpose of an orthogonal matrix is the same as its
inverse.
FP3
Y
2d Co-ordinate Geometry – equations of straight
lines
All students should learn ax+by+c=0, y=mx+c and y-y1=m(x-x1) in module C1. It
𝑦−𝑦
𝑥−𝑥
is quite likely that they will also know 𝑦 −𝑦1 = 𝑥 −𝑥1 although this is not
C1
N
2
1
2
1
compulsory. Nonetheless, this should be straightforward to any maths
student. They should know the gradient formula but only students offering
STEP or AEA are likely to have connected this with tanθ.
Intersections of lines and perpendicular lines
All students should have little trouble with intersections and the condition that
the gradient of a perpendicular line is the negative reciprocal of the original
gradient is known to all students from module C1 and is used often throughout
A level in calculating the equations of normal to curves. All students study
module C3 and consequently should have little to fear from the use of cotθ.
C1/C3
N
Midpoints and distances
The formulae for midpoint and distance between two points should be known
to all students from modules C2. However the formula for distance from a
point to a line will likely be new to all students.
C2
N
Circles
The circle theorem about tangents should be known to all GCSE students. The
equation of a circle, intersection of line and circles and the use of completing
the square to find the centre and radius of a circle should be routine for all C2
students. They are unlikely to have examined the intersection of two circles.
The parametric form will come as little surprise to students of FP3 who know
the parametric equations of the ellipse. All students study parametric
equations on module C4.
C2/FP3/C4
N
Two circle compound motion
This will be new to all students. In general, unless a student has studied AEA or N/A
STEP, they are likely to have poor geometric intuition so this may well prove
difficult for them
N/A
The parabola
The form y2=4ax and the focus directrix property should be known to all FP1
students. It is rare that the parabola will be moved away from the origin or
found in a different orientation, though.
FP1
Y
Tangents to the parabola
This style of question will be quite unfamiliar. At A level, students are normally
given a point on the curve at which to calculate a tangent.
N/A
N/A
The reflection property of the parabola
It would not be uncommon for a student of FP1 to have encountered this but it
is not compulsory
N/A
N/A
Parametric form of the parabola
This will be known to students of FP1 and should be of little difficulty to all
students from C4 parametric equations
FP1/C4
Y
The ellipse
Students of FP3 should know the Cartesian and parametric equations of the
ellipse, the focus-directrix property and should have little trouble finding
tangents. It is not uncommon for them to know that the sum of the distances
of a point on the ellipse to the two foci is constant (however, students do find
locus problems very difficult). They are very unlikely to know the reflection
property.
FP3
Y
The hyperbola
Students of FP3 should know the Cartesian and parametric equations of the
hyperbola, the focus-directrix property and should have little trouble finding
tangents. They should also know the equations of the asymptotes. They are
very unlikely to know the reflection property or to have calculated tangents for
a point not on the hyperbola. Students of FP1 study the rectangular hyperbola
– they will know xy=c2 and the corresponding parametric equations but will
not have considered the significance of the asymptotes being at right angles
FP3/FP1
Y
Quadratic curves
It is unlikely that any student will have considered the equations of ellipses and
hyperbolae not in their standard position and orientation
N/A
N/A