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ES193 - Engineering Mathematics - Week 6 Practise Questions and Solutions - University of Warwick
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Questions and Solutions for week 6 of the ES193 Engineering Mathematics module for the Engineering course.
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School of Engineering, University of WarwickES193 Engineering Mathematics
Briefing Sheet:
Topic: MATRICES
1. Motivation
This week we start to study matrices, part of a subject known as linear algebra. Matrices are
widely used in engineering where equations are set up to describe a large engineering problem by
a series of smaller problems. An example of this is where the stresses and displacements in a beam
structure might be calculated by first finding equations to describe each beam in the structure,
before solving these together.
2. Key Concepts
This week the key concepts are:
• definition of a matrix
• shape of a matrix
• addition, subtraction and multiplication of matrices
• transpose and symmetry properties
• inverse of a matrix.
3. Learning Outcomes
By the end of this week’s work you should be able to:
• add and subtract matrices
• find the transpose of a matrix
• multiply matrices
• find the inverse of a matrix.
4. This Week’s Reading
This week we will tackle Chapter 12 which effectively introduces the concept of a matrix and details
matrix arithmetic. The work that we do should complement the work that you are currently doing
in connection with modelling and system dynamics, especially in the computational laboratories
where you are using MATLAB to manipulate matrices. Note that we have previously already
covered one aspect - the determinant - of a matrix. In that we work our motivation was to obtain
a simple way to evaluate the vector product of two vectors. Now, we will use the determinant as
part of the technique for finding the inverse of a matrix, the final objective of this week’s work.
Next week, we will see how the inverse of a matrix can be used to solve systems of simultaneous
(linear) equations.
As in previous weeks, I am suggesting certain questions from the block exercises which serve to
help you learn. You should do as many of these as you feel necessary to be confident that you
have understood the topic. Matrix arithmetic can be very laborious in terms of repeated numerical
calculations (hence the existence of computer packages like MATLAB), so don’t get too bogged
down with the sums - it is understanding the principles which is important. Finally use the
revision questions in §6 as the real test of whether you have understood the topic and can answer
examination-type questions.
1
, 5. Example Problems
Chapter 12
Pages 523-533: BLOCK 1: Introduction to matrices
(Definition and nomenclature - e.g. C13 is the element of Row 1 in Column 3;
main (or leading) diagonal of a square matrix; identity matrix; transpose of a
matrix; matrix arithmetic - addition, subtraction and scaling)
Exercises: Page 524, Q1; Page 526, Q1; Page 528, Q1; Page 531, Q2, Q3; Page 532
(End-of-block), Q1, Q2, Q3, Q9, Q13
Pages 535-543: BLOCK 2: Multiplication of matrices
(Condition (sizes) for two matrices to be multiplied together; premultiplication,
postmultiplication.)
Exercises: Page 535, Q1, Q3; Page 541, Q2, Q3, Q6, Q7 (noting that ABC 6= BCA...
order of multiplication matters!); Page 542 (End-of-block), Q10, Q13 (Noting
that (AB)T = BT AT in general)
BLOCK 3: This was covered earlier
Its contents - cofactors and determinants - will now be used to find the inverse
of a matrix.
Pages 563-572: BLOCK 4: The inverse of a matrix
(The notation A−1 for the inverse of matrix A (this follows the notation f −1 (x)
being the inverse of a function f (x)); inverse of 2 × 2 matrix; the general method
to find the inverse of a matrix - note that you could be expected, routinely, to
find the inverse of a 3×3 matrix but not larger. On Page 542, the rules for matrix
inversion: whether the transpose is taken before or after finding the matrix of
cofactors does not matter.)
Exercises: Page 566, Q1, Q2 (a), (c) and (d) only; Page 568, Q1 (a) and (b) only; Page
569-570 (End-of-block) Q1, Q2, Q4, Q9, Q10 (b) only.
Pages 572-599: BLOCK 5: Computer graphics
(i.e. do not cover §5.10)
(Matrices as linear transformations: scaling, rotation, reflection, shearing, trans-
lation.)
Exercises: Page 575, Q1; Page 577, Q1; Page 580, Q1, Q3; Page 584, Q1, Q2; Page
585, Q1; Page 588, Q3.
6. Revision Questions for Tutorials
At the completion of the above learning, now attempt each of the following revision questions on
matrix manipulation and applications. Your tutor will expect you to be familiar with (either have
done or attempted) these questions because they will be raised at your relevant tutorial.
Q1 (*) Three matrices are defined as follows:
1 2
2 1 −1 2
A= B = 3 1 C=
3 4 2 −1
1 0
2
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