Transcript FP1 - Chapter 4 - Matrix Algebra

FP1: Chapter 4 Matrix Algebra
Dr J Frost ([email protected])
Last modified: 10th February 2014
Introduction
A matrix (plural: matrices) is simply an ‘array’ of numbers, e.g.
On a simple level, a matrix is simply a way to organise values into rows and columns,
and represent these multiple values as a single structure.
But the power of matrices comes from being able to multiply matrices by vectors
and matrices by matrices and ‘invert’ them: we can:
1. represent linear transformations using matrices (e.g. rotations, reflections
and enlargements)
2. Use them to solve linear simultaneous equations.
The first of these means matrices are particularly useful in 3D
graphics/animation, since they allow us to rotate the camera and project 3D
data onto a 2D viewing plane. But matrices are used everywhere, including
robotics, computer vision, optimisation, classical and quantum mechanics,
electromagnetism, optics, graph theory, statistics, ...
Using matrices to represent data
This is a scene from the film Good Will?Hunting.
Maths professor Lambeau poses a difficult* problem for his graduate students from
algebraic graph theory, the first part asking for a matrix representation of this graph.
Matt Damon anonymously solves the problem while on a cleaning shift.
In an adjacency matrix, the
number in the ith row and jth
column is the number of
edges directly connecting
node (i.e. dot) i to dot j
?
* It really isn’t.
Using matrices to represent data
In my 4th year undergraduate dissertation, I used matrices to help ‘learn’ mark schemes from
GCSE biology scripts. Matrix algebra helped me to initially determine how words (and more
complex semantic information) tended to occur together with other words.
Using matrices to represent data
In S1, you learnt how we could extend the idea of variance (i.e. how much a variable
varies) to that of covariance (i.e. how much two variables vary with each other).
A covariance matrix allows us to store the covariance between variables all in one
convenient structure.
This allows us for example to extend a Normal Distribution from one dimension (i.e.
involving just one variable) to multiple dimensions.
1D
2D
Matrix Algebra
Matrix Fundamentals
Understand the dimensions of a matrix, and operations on
matrices, such as addition, scalar multiplication and matrix
multiplication.
Matrix Fundamentals
#1 Dimensions of Matrices
The dimension of a matrix is its size, in terms of its number of rows and columns.
Matrix
Dimensions
23
31
?
13
?
Matrix Fundamentals
#2 Notation/Names for Matrices
A matrix can have square or curvy brackets*.
Matrix
Column Vector
Row Vector
(The vector you know
and love)
So a matrix with one column is simply a vector in the usual sense.
* The textbook only uses curvy.
Matrix Fundamentals
#3 Variables for Matrices
If we wish a variable to represent a matrix, we use bold, capital letters.
Matrix Fundamentals
#4 Elements
Each value within a matrix is known as an element.
If A is a matrix, then we can refer to the element in the ith row and jth
column as aij.
c12 = 3?
c21 = 4?
c13 = -7?
Note that you do not need to know this notation for referencing elements
for the purposes of FP exams, but it’s worthwhile knowing.
Matrix Fundamentals
#5 Adding/Subtracting Matrices
Simply add/subtract the corresponding methods of each matrix.
They must be of the same dimension.
?
?
Matrix Fundamentals
#6 Scalar Multiplication
A scalar is a number which can ‘scale’ the elements inside a matrix/vector.
1
2
3
?
?
?
Matrix Fundamentals
#7 Matrix Multiplication
This is where things get slightly more complicated...
Now repeat for the next row of the left matrix...
1
2
7
0 3 -2
8 4 3
-1 0 2
5
1
0
8
1
7
3
-3
-11
42
16
61
50
8
We start with this row and column, and sum the products of each pair.
(1 x 5) + (0 x 1) + (3 x 0) + (-2 x 8) = -11
You will see in C4 that this is known as finding the “dot/scalar product”
of the two vectors.
Matrix Fundamentals
#7 Matrix Multiplication
Now you have a go...
=
?
?
?
?
Matrix Fundamentals
#7 Matrix Multiplication
Matrix multiplications are not always valid: the dimensions have to agree.
Dimensions of A
Dimension of B
Dimensions of AB (if valid)
23
34
24
13
23
Not valid.
62
24
64
13
31
11
75
75
Not valid.
10  10
10  9
10  9
33
33
33
?
?
?
?
?
?
?
Note that only square matrices (i.e. same width as height) can be raised to a power.
Exercise 4C
Q1, 3, 5, 6, 8, 9
Matrix Algebra
Linear Transformations
Appreciate what linear transformations are, and how we can use
matrices to represent them.
Position vectors
You should be familiar with the difference between a vector and a point:
Point:
Vector:
? in space.
represents a position
represents a movement.
?
A position vector allows us to treat a point as a vector.
A position vector [x y] represents the movement from the origin (0,0) to the point (x,y).
y
Position
vector [5 3]
Point (5,3)
x
For the remainder of the chapter, when I say ‘point’, I really mean ‘position vector’.
Linear Functions
From GCSE, you are used to a linear function looking like the equation of a straight line:
f(x) = ax + b
More generally, if we had multiple inputs, a linear function is:
f(x1, x2, ... , xn) = a0 + a1x1 + a2x2 + ... + anxn
When the inputs and outputs of the function are vectors however, then we have a
linear map/transformation if:
f(ax) = af(x)
f(x + y) = f(x) + f(y)
The first condition ensures that if we scale the original point x, then this scales the
transformed point too. The second says that if we add two vectors, then resulting
vector will be the same then if we transformed each of the vectors individually then
added them – we say that the transformation ‘preserves vector addition and scalar
multiplication’.
Linear Transformations
Suppose that 0 is the 0-vector, i.e. The position vector (0,0) in 2D, (0,0,0) in 3D, etc.
Then for a linear transformation f:
f(0) = 0?
This follows by letting the scalar a = 0 in f(ax) = af(x).
This means that the origin is unaffected by a linear transformation.
Can you express the following transformations as functions?
If linear, prove it, otherwise find a counterexample.
Example 1
Reflects a 2D point in the y-axis.
?
Proof of
linearity:
?
Linear Transformations
Example 2
Rotates a 2D point 90 clockwise about the origin.
?
Linear?
Example 3
Yes (proof similar
? to before)
Enlarges by a scale factor of 3 with the origin as the centre of
enlargement.
?
Linear?
Yes. Proof:
?
Linear Transformations
Example 4
Projects a 3D coordinate onto the xy-plane, almost as if you were
observing a 3D point where your viewing window/eyes are the xy-plane.
?
Linear?
Yes (proof left?as exercise)
This is useful in 3D graphics, because you need to turn 3D data into a 2D
image by projecting onto a viewing plane.
Example 5
Translate a 2D point 1 unit to the right.
?
Linear?
No! The origin is?not preserved.
Linear Transformations
As we have seen, the following transformations are linear and not linear:
Linear
Non-Linear
•Reflection (where line of
reflection goes through the
origin)
•Rotation (with centre at the
origin)
•Enlargement (with centre at
the origin)
•Translation.
A linear transformation in general is when each output is a linear
combination of the inputs, e.g:
While a transformation can create or destroy dimensions, in FP1 we’ll just be
dealing which transformation which transform 2D points to 2D points.
Linear Transformations
The following examples are NOT linear transformations:
(can you prove it?)
𝑥
𝑥𝑦
𝑓 𝑦 = 𝑦
𝑥
𝑥2
𝑓 𝑦 =
𝑦
Affine Transformations
(Not mentioned in FP1)
Affine transformations extend linear transformations by allowing a constant
term. This hence allows us to represent translations.
Linear
𝑥
2𝑥
𝑓 𝑦 =
𝑥−𝑦
Affine
𝑥
2𝑥 + 𝟏
𝑓 𝑦 =
𝑥−𝑦
𝑥
𝑥+𝟏
𝑓 𝑦 =
𝑦−𝟐
Using Matrices for Linear Transformations
Matrices allow us to represent all possible linear transformations.
Reflection in 𝑦-axis:
𝑥
−𝑥
𝑓 𝑦 = 𝑦
−𝟏 𝟎
?
𝟎 𝟏
𝑥
−𝑥
𝑦 = 𝑦
𝟎 𝟏
?
−𝟏 𝟎
𝑥
𝑦
𝑦 = −𝑥
Rotation 90° clockwise about the origin.
𝑥
𝑦
𝑓 𝑦 =
−𝑥
Using Matrices for Linear Transformations
Matrices allow us to represent all possible linear transformations.
𝑥
2𝑦 + 𝑥
𝑓 𝑦 =
3𝑥
𝑥
−2𝑦
𝑓 𝑦 =
3𝑥 + 𝑦
2𝑦 + 𝑥
𝟏?𝟐 𝑥
=
𝑦
𝟑 𝟎
3𝑥
−2𝑦
𝟎 −𝟐 𝑥
=
?
𝑦
3𝑥 + 𝑦
𝟑 𝟏
𝑥
2𝑦 − 𝑥
𝑓 𝑦 =
3𝑦 + 𝑥
𝑥
0
𝑓 𝑦 =
𝑥
−𝟏 𝟐
?
𝟏 𝟑
𝟎 𝟎
?
𝟏 𝟎
𝑥
2𝑦 − 𝑥
𝑦 = 3𝑦 + 𝑥
𝑥
0
𝑦 = 𝑥
Basis Vectors and Vector Spaces
(Not mentioned in FP1)
Basic vectors are usually (but not necessarily) the unit vectors along each axis.
A vector space is all possible points that can be obtained by some linear combination
of the basis vectors.
Example: 2D space
Basic vectors:
Vector space:
1
0
0
1
𝑥
𝑥
1
0
All position vectors 𝑦 since 𝑦 = 𝑥
+𝑦
0
1
Intuitively this is obvious: any point in 2D space can be obtained by some
movement along the 𝑥 axis and some movement along the 𝑦 axis.
Our basis vectors could have been any non-parallel vectors.
Using Matrices for Linear Transformations
There’s a groovy trick that allows us to work out what matrix to use for a
particular transformation. Observe:
𝑎
𝑐
𝑎
1
=
𝑐
0
𝑏
𝑑
𝑎
𝑐
0
𝑏
=
1
𝑑
𝑏
𝑑
Thus, each column of the matrix tells us how the unit vectors are
transformed. These transformed basis vectors form the new vector space.
y
Example: Reflection in 𝑦-axis.
y
First column is first basis
vector transformed.
𝑇=
−1 0
?
0 1
0
1
0
1
1
0
x
−1
0
x
Using Matrices for Linear Transformations
Rotation 90° anticlockwise
about the origin.
0 −1
𝑇=
?
1 0
0
1
0
1
1
0
x
−1
0
Rotation 𝜃 anticlockwise about
the origin.
x
𝑇=
cos 𝜃
sin 𝜃
− sin 𝜃
? cos 𝜃
−sin 𝜃
?
cos 𝜃
0
1
1
0
cos 𝜃
?
sin 𝜃
x
𝜃
x
Rotation for multiples of 45°
In the FP1 specification, you are only required to identify rotations for multiples of 45°,
although the rotation matrix for 𝜃 anticlockwise in general is in your formula booklet.
1
Rotation of 45°
anticlockwise
2
1
2
Rotation of 45°
clockwise
Rotation of 135°
anticlockwise
1
−
cos 𝜃
sin 𝜃
1
1
2
?=
2
1 1
2 −1 1
?
1 −1 −1
?
2 1 −1
1 1 −1
2 1 1
− sin 𝜃
cos 𝜃
Bro Tip: To remember
these without having
to plug in your 𝜃 in
the above matrix, just
1
remember the 2
scaling, and then just
visually think how
each basis vector is
transformed to get
the signs of the 1s
correct.
Using Matrices for Linear Transformations
Enlargement by scale factor 2
about the origin.
0
?
2
2 0
𝑇=
?
0 2
0
1
1
0
2
?
0
x
No change.
This is known as
the identity
matrix, and is
denoted by 𝑰.
0
1
1
0
x
𝑇=
x
1 0
?
0 1
0
1
1
0
x
A quick digression…
(Not mentioned in FP1)
An identity element in general is a special type of element on a set (e.g. the set of
integers, or real numbers, or matrices) with respect to a binary operator on the set.
An element 𝒆 is an identity element if for all 𝒂 in the set and some operator ⨂ then:
𝒂⨂𝒆 = 𝒂
Set
Operation
Identity Element
ℝ
+
0 (since 𝑎 + 0 = 𝑎)
ℝ
×
1 (since 𝑎 × 1 = 𝑎)
𝑛 × 𝑛 square
matrices
Matrix
multiplication
?
?
Sets
∪
𝑰𝒏 (since 𝑰𝒏 𝑨 = 𝑨)
1 0 0 ?
(e.g 𝐼3 = 0 1 0 , but you tend to just see 𝑰)
0 0 1
∅ (since 𝐴 ∪ ∅ = 𝐴) ?
Boolean
Algebra
∧
⊤ (since 𝑎 ∧ ⊤ = 𝑎)
𝑛×𝑚
matrices
Matrix
addition
The “0-matrix” 𝟎.
?
+
The “0-vector” 𝟎.
?
Vectors
?
Using Matrices for Linear Transformations
Reflection in the line 𝑦 = tan 𝜃 𝑥
0
1
𝜃
1
0
𝑇=
180 − 2𝜃
𝜃
x
x
cos 2𝜃
?
sin 2𝜃
Since for any line of length 𝑚 and inclination 𝜃,
the 𝑥 component is 𝑚 cos 𝜃 and 𝑦 component
𝑚 sin 𝜃, then gradient is consequently tan 𝜃,
i.e. 𝜃 must be the angle between the 𝑥 axis
and line of reflection.
sin 2𝜃
?
− cos 2𝜃
We can see from the
diagram above that when
1
we reflect
we can
0
form the following
triangle:
Thus the vector gets transformed to
− cos 180 − 2𝜃
cos 2𝜃
=
sin 180 − 2𝜃
sin 2𝜃
1
180 − 2𝜃
We can do
something similar
0
to transform
1
Exercise 4E
1
Describe the transformations
represented by the matrix:
4 Find the matrix that represents these
transformations.
1 0
Reflection?in 𝒙-axis.
0 −1
0 −1
b)
Rotation 𝟗𝟎° anticlockwise
1 0
?
a)
c)
3
0 1
−1 0
about 𝟎, 𝟎 .
Rotation 𝟗𝟎° clockwise
?
about 𝟎, 𝟎 .
And for these:
a)
1
2
1
− 2
4 0
b)
0 4
c)
1
2
−1
−1
1
2
1
2
5
Rotation 𝟒𝟓° clockwise
?
about 𝟎, 𝟎 .
Enlargement scale
?
factor 4, centre
1
−1
a) Rotation of 90° clockwise about 0,0 .
0 1
−1 ?0
b) Reflection in the 𝑥-axis.
1 0
?
0 −1
c) Enlargement centre 0,0 scale factor 2.
2 0
0 ?2
𝟎, 𝟎 .
Rotation 𝟐𝟐𝟓°
?
anticlockwise about 𝟎, 𝟎 .
Find the matrix that represents these
transformations.
a) Enlargement scale factor -4 centre (0,0).
−4
0
0
−4
?
b) Reflection in the line 𝑦 = 𝑥.
0 1
1 0
?
c) Rotation about 0,0 of 135° clockwise.
1 −1
2 1
−1
−1
?
Combined Transformations
We know that for a position vector 𝒙 and a matrix 𝑨 representing some
transformation, then 𝑨𝒙 is the transformed point.
If we wanted to apply a transformation represented by a matrix 𝑨 followed by
another represented by 𝑩, what transformation matrix do we use to represent the
combined transformation?
?
𝑩𝑨
This is because to apply the effect of 𝐴 followed by 𝐵,
we have:
𝐵 𝐴𝑥 = 𝐵𝐴𝑥 = 𝐵𝐴 𝑥
(because matrix multiplication is ‘associative’*)
Bro Tip: Ensure that you put these matrices in the right
order – the first that gets applied is on the right!
* A binary operator ⨂ is associative if 𝑎⨂ 𝑏⨂𝑐 = (𝑎⨂𝑏)⨂𝑐,
i.e. when we multiply matrices, the order in which we multiply
them doesn’t matter.
Combined Transformations
Represent as a single matrix the transformation representing a reflection in the line
𝑦 = 𝑥 followed by a stretch on the 𝑥 axis by a factor of 4.
4 0
0 ?1
0 1
? = 0?4
1 0
1 0
Represent as a single matrix the transformation representing a rotation 90°
anticlockwise about the point 0,0 followed by a reflection in the line 𝑦 = 𝑥.
0 1
1 ?0
0 −1
? = 1 ?0
1 0
0 −1
What single transformation is this?
Reflection in?the 𝒙 axis.
Exercise 4F
Page 93
Q1, 3, 5, 7
Matrix Inverses
In maths, we are used to functions having an inverse.
𝑓 𝑥 =𝑥+3
𝑓 𝑥 = 2𝑥
𝑓 −1 𝑥 = 𝑥 − 3
𝑥
−1
𝑓 𝑥 =
2
It would seem logical to have some inverse of multiplying something by a
matrix, so that we can represent ‘undoing’ the transformation (e.g. the inverse
of the matrix representing 90° rotation clockwise would be a matrix
representing a rotation 90° anticlockwise).
! If 𝑨 =
𝑎
𝑐
1
𝑏
𝑑
then 𝑨−1 =
𝑎𝑑−𝑏𝑐 −𝑐
𝑑
−𝑏
𝑎
• 𝑨−𝟏 is the ‘inverse’ of 𝑨, so that if 𝑨𝒙 = 𝒚, 𝑨−1 𝒚 = 𝒙
• 𝑨𝑨−𝟏 = 𝑨−𝟏 𝑨 = 𝑰 since the effect of a transformation
followed by its inverse has no effect.
Determinants
𝑎 𝑏
is
𝑐 𝑑
det 𝑨 = 𝑎𝑑 − 𝑏𝑐
! The determinant of a matrix 𝑨 =
So 𝑨−1 =
1
det 𝐴
𝑑
−𝑐
−𝑏
𝑎
! If 𝑑𝑒𝑡 𝑨 = 0, then 𝑨 is a singular matrix and 𝑨−1 does not exist
(since we can’t divide by 0).
! If det 𝑨 ≠ 0, then 𝑨 is a non-singular matrix and 𝑨−1 exists.
A
det(A)
1 0
0 1
1 2
3 4
0
3
−1 −4
10 −2
4 −1
1
?
-2
?
3
?
-2
?
It’s worthwhile reflecting
on the significance of the
determinant: in the
context of
transformations, it
‘determines’ whether we
can ‘undo’ a
transformation. And
later, we’ll see that it can
tell us whether a series
of simultaneous
equations are solvable.
Practicing the Inverse
1
𝑎𝑑 − 𝑏𝑐
Divide by
determinant.
𝑎
𝑐
−
−
𝑏
𝑑
Swap NW-SE
elements.
Click to
Broinverse
Make SW-NE
elements negative.
Test Your Understanding
0
2
−1
1 1 0
?
=
2 0 1
1
3
2
4
−1
1 −4 2
=
?
2 3 −1
7
1
2
−3
2
0
−1
1 3 2
=
?
23 1 −7
4 𝑝+2
For what value of 𝑝 is
singular? Given 𝑝 is not
−1 3 − 𝑝
this value, find the inverse.
𝒑=
𝟏𝟒
?
𝟑
𝟏
𝟏𝟒−𝟑𝒑
𝟑−𝒑 − 𝒑+𝟐
?
𝟏
𝟒
Using 𝐴𝐴−1 = 𝐴−1 𝐴 = 𝐼
A matrix multiplied by its inverse is the identity matrix.
If 𝑨 and 𝑩 are non-singular matrices, prove that 𝑨𝑩
If 𝐶 = 𝐴𝐵
−1
then 𝐴𝐵 𝐶 = 𝐼
𝐴−1 𝐴𝐵𝐶 = 𝐴−1 𝐼
𝐴−1 𝐴 𝐵𝐶 = 𝐴−1 𝐼
𝐵𝐶 = 𝐴−1?
𝐵 −1 𝐵𝐶 = 𝐵 −1 𝐴−1
𝐵 −1 𝐵 𝐶 = 𝐵 −1 𝐴−1
𝐶 = 𝐵 −1 𝐴−1
−𝟏
= 𝑩−𝟏 𝑨−𝟏
The general strategy for
solving these kinds of
equations is to multiply
whatever’s on the front of
the multiplication by its
inverse, to ‘cancel it out’.
If 𝑨 and 𝑩 are 𝟐 × 𝟐 non-singular matrices such that 𝑩𝑨𝑩 = 𝑰. Prove that
𝑨 = 𝑩−𝟏 𝑩−𝟏.
𝐵𝐴𝐵 = 𝐼
𝐵 −1 𝐵𝐴𝐵 = 𝐵−1 𝐼
?
𝐴𝐵 = 𝐵 −1
𝐴𝐵𝐵 −1 = 𝐵 −1 𝐵 −1
𝐴 = 𝐵 −1 𝐵−1
Exercise 4G
Page 97
Q1a, c, e
2a, c
4, 6, 8, 10, 11
Inverse matrices represent inverse transformations
! Suppose 𝒙 and 𝒚 are column vectors. Then if 𝑨𝒙 = 𝒚, then 𝒙 = 𝑨−𝟏 𝒚.
The inverse matrix therefore allows us to retrieve the original point/position vector
before a transformation.
The triangle 𝑻 has vertices at 𝑨, 𝑩 and 𝑪. The matrix 𝑴 = 4 −1
3 1
transforms 𝑻 to the triangle 𝑻′ with vertices at 𝟒, 𝟑 , 𝟒, 𝟏𝟎 and −𝟒, −𝟑 .
Sketch the two triangles, and hence show that 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑇 ′ =
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑇 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑑𝑒𝑡(𝑴)
𝑴−𝟏 =
1
7
1 1
−3 4
1 1 1 4
𝐴=
=
7 −3 4 3
1 1 1 4
𝐵=
=
7 −3 4 10
1 1 1 −4
𝐶=
=
7 −3 4 −3
(4,10)
(2,4)
(4,3)
1
0
2
4
−1
0
-4
?
-1
4
1
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑻 = 𝟒
𝑨𝒓𝒆𝒂 𝒐𝒇 𝑻′ = 𝟐𝟖
𝒅𝒆𝒕 𝑴 = 𝟕
So the determinant was the scale factor
of the area.
Area scale factor
We saw in this example that:
! Area of image = Area of object × det 𝑴
i.e. the determinant tells us how the area is scaled under the transformation
matrix 𝑴.
Area of Object
4
3
9
1
Transformation
Matrix
1 2
3 4
2 0
9 4
5
3
−2 −1
−5 2
−4 −2
Area of Image
4×2=8
?
3 × 8 = 24
?
9×1=9
?
1 × 18 = 18
?
Exam Question
Edexcel FP1 - Jan 2011
4
?
1 3 2
?
4 1 2
72 ÷ ?
4 = 18
Note that we’re given the points on the image! So use inverse transformation.
1 3 2
1 3 2
1 3 2
8
14
11
0
2
12
=
=
=
?
4 1 2
4 1 2
4 1 2
16
10
5
4
2
4
So points are 2,2 , 14,10 , 11,5 .
Exercise 4I
2 −1
is used to transform the rectangle 𝑅 with
4 3
vertices at the points 0,0 , 0,1 , 4,1 and 4,0 .
a) Find the coordinates of the vertices of the image of 𝑅.
b) Calculate the area of the image of 𝑅.
0,0 , −1,3 , 7,19 , 8,16
?
Area = 40
1
The matrix 𝐴 =
4
2 −4
1 2
1 2
𝑄=
𝑅=
3 1
−1 4
2 1
2
A rectangle of area 5cm is transformed by the matrix 𝑋. Find the area of
the image of the rectangle when 𝑋 is:
a) 70 c) 15 ? e) 90
a) 𝑃
c) 𝑅
e) 𝑄𝑅
5
The triangle 𝑇 has area 6cm2 and is transformed by the matrix 𝐀 =
𝑎
3
where 𝑎 is a constant, into triangle 𝑇′.
3 𝑎+2
a) Find det(𝑨) in terms of 𝑎.
b) Given that the area of 𝑇′ is 36cm2, find the possible values of 𝑎.
𝑃=
a) 𝑎2 = 2𝑎 − 9
b) −5, −3, 1 ?
𝑜𝑟 3
Frost Life StoriesTM
In the game Assassin’s Creed II, you encounter a variety of concentric ring picture
puzzles, which upon successfully completing, you unlock a segment of a secret video.
Rings are connected in pairs, and must be rotated together in their pairs. The aim is to
form a complete picture. Different possible pairs can be selected, for example, where
there just 3 rings, you could rotate A and B together, B and C together or C and A
together.
Only certain pairings are available.
Because I’m a massive geek, I formed
simultaneous equations and used a
matrix inverse to solve them, which
therefore told me how many times to
rotate each pair.
We’ll see how we can do this.
Using Matrices For Simultaneous Equations
Turn the following simultaneous equations into a matrix:
2𝑥 − 3𝑦 = 5
−5𝑥 + 6𝑦 = −8
2 −3
−5 6
𝑥
5
=
𝑦 ? −8
Hence, use a matrix inverse to solve these simultaneous equations.
Let 𝑴 =
𝟐
−𝟓
𝟏 𝟔
−𝟑
. Then 𝑴−𝟏 = − 𝟑
𝟔
𝟓
𝟑
𝟐
𝒙
𝟓
𝑴 𝒚 =
−𝟖
𝒙
𝟓
⇒ 𝑴−𝟏 𝑴 𝒚 ?
= 𝑴−𝟏
−𝟖
𝒙
𝟓
⇒ 𝒚 = 𝑴−𝟏
−𝟖
−𝟐
=
−𝟑
So 𝒙 = −𝟐 and 𝒚 = −𝟑.
Test your understanding
Q Solve (using matrices) the simultaneous equations:
2𝑥 + 𝑦 = 9
3𝑥 − 𝑦 = 16
𝟐 𝟏
𝟑 −𝟏
𝒙
𝟗
=
𝒚
𝟏𝟔
𝟏 −𝟏
𝒙
𝟐 𝟏 −𝟏 𝟗
=−
𝒚 = 𝟑 −𝟏
𝟏𝟔
𝟓 −𝟑
𝟏 −𝟐𝟓
𝟓
=−
=
−𝟏
𝟓 𝟓
−𝟏 𝟗
𝟐? 𝟏𝟔
So 𝒙 = 𝟓, 𝒚 = −𝟏.
Q
Explain why the following simultaneous equations have no solution:
2𝑥 + 𝑦 = 5
4𝑥 + 2𝑦 = 6
The matrix
𝟐
𝟒
𝟏
has no inverse because
? the determinant is 0.
𝟐
Exercise 4J
Use inverse matrices to solve the following simultaneous equations.
1a
7𝑥 + 3𝑦 = 6
−5𝑥 − 2𝑦 = −5
1b
4𝑥 − 𝑦 = −1
4𝑥 − 𝑦 = 11
𝒙 = 𝟑, 𝒚 ?= −𝟓
𝒙 = 𝟎. 𝟓, 𝒚 = 𝟑
?
𝒙 = 𝟐, 𝒚 = −𝟑
2a
?
𝒙 = −𝟏, 𝒚 = 𝟒
2b
?