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Unit 39
Matrices
Presentation 1
Matrix Additional and Subtraction
Presentation 2
Scalar Multiplication
Presentation 3
Matrix Multiplication 1
Presentation 4
Matrix Multiplication 2
Presentation 5
Determinants
Presentation 6
Inverse Matrices
Presentation 7
Solving Equations
Presentation 8
Geometrical Transformations
Presentation 9
Geometric Transformations: Example
Unit 39
39.1 Matrix Additional and
Subtraction
If a matrix has m rows and n columns, we say that its dimensions
are m x n.
For example
is a 2 ?x 2 matrix
is a 2 ?x 3 matrix
You can only add and subtract matrices with the same
dimensions; you do this by adding and subtracting their
corresponding elements.
Example 1
(a)
(b)
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Example 2
If
what are the values of a, b, c
and d?
Solution
Subtracting gives
Hence
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Unit 39
39.2 Scalar Multiplication
For scalar multiplication, you multiply each element of the matrix
by the scalar (number) so
Example
If
then
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Unit 39
39.3 Matrix Multiplication 1
You can multiply two matrices, A and B, together and write
only if the number of columns of A = number of rows of B; that is,
if A has dimension m x n and B has dimension n x k, then the
resulting matrix, C, has dimensions m x k.
To find, C, we multiply corresponding elements of each row of A
by elements of each column of B and add. The following
examples show you how the calculation is done.
Example
If
and
, then A is a 2 x 2 matrix and B is
a 2 x 1 matrix, so C = AB is defined and is a 2 x 1 matrix, given by:
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Unit 39
39.4 Matrix Multiplication 2
Here we show a matrix multiplication that is not commutative
Consider
and
First we calculate AB.
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Here we consider a matrix multiplication that is not commutative
Consider
and
And now for BA.
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Is AB = BA? No
Hence matrix multiplication is NOT commutative
Unit 39
39.5 Determinants
For a 2 x 2 square matrix
number defined by
its determinant is the
Example 1
What is detA if
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Solution
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For a 2 x 2 square matrix
number defined by
its determinant is the
Example 2
If
what is the value of x that would make
detM = 0 ?
Solution
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A matrix, M, for which detM = 0 is called a
singular matrix.
Unit 39
39.6 Inverse Matrices
For a 2 x 2 matrix, M, its inverse
, is defined by
where
You can always find the inverse of M if it is non-singular, that is
. For
Example
If
find
Solution
and verify that
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Hence
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Unit 39
39.7 Solving Equations
You can write the simultaneous equation
In the form
when
You can solve for X by multiplying
by
This gives
or
So we first need to find
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and
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Hence
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Unit 39
39.8 Geometrical
Transformations
You can use matrices to describe transformations. We write
where
is transformed into
Lets look at the common transformations
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Unit 39
39.9 Geometric Transformations:
Example
Example
A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-1, 4) is
mapped onto triangle Xʹ Yʹ Zʹ by a transformation
(a) Calculate the coordinates of the vertices of triangle Xʹ Yʹ Zʹ
Solution
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i.e.
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i.e.
i.e.
Example
A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is
mapped onto triangle Xʹ Yʹ Zʹ by a transformation
(b) A matrix
maps triangle Xʹ Yʹ Zʹ onto triangle
Xʹʹ Yʹʹ Zʹʹ. Determine the 2 x 2 matrix, Q, which maps triangle
XYZ onto Xʹʹ Yʹʹ Zʹʹ.
Solution
Xʹʹ = NXʹ = NMX so Xʹʹ = QX where
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Example
A triangle, XYZ, with coordinates X (4, 5), Y(-3, 2) and Z(-4, 4) is
mapped onto triangle Xʹ Yʹ Zʹ by a transformation
(c) Show that the matrix which maps triangle Xʹʹ Yʹʹ Zʹʹ back onto
XYZ is equal to Q.
Solution
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so QXʹʹ = X and similarly QYʹʹ = Y and QZʹʹ = Z
Thus Q maps Xʹʹ Yʹʹ Zʹʹ back to XYZ