Post on 17-Jan-2016
Unit 39Matrices
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 3939.1 Matrix Additional and
Subtraction
If a matrix has m rows and n columns, we say that its dimensions are m x n.
For exampleis 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.
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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 3939.2 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 3939.3 Matrix Multiplication 1
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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 3939.4 Matrix Multiplication 2
Here we show a matrix multiplication that is not commutative
Consider and
First we calculate AB.
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Is AB = BA? NoHence matrix multiplication is NOT commutative
Here we consider a matrix multiplication that is not commutative
Consider and
And now for BA.
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Unit 3939.5 Determinants
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For a 2 x 2 square matrix its determinant is the number defined by
Example 1
What is detA if ?
Solution
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For a 2 x 2 square matrix its determinant is the number defined by
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 3939.6 Inverse Matrices
For a 2 x 2 matrix, M, its inverse , is defined by
You can always find the inverse of M if it is non-singular, that is . For
Example
If find and verify that
Solution
Hence
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where
Unit 3939.7 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 . Now
and
Hence
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Unit 3939.8 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 3939.9 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.
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
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(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
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
so QXʹʹ = X and similarly QYʹʹ = Y and QZʹʹ = Z
Thus Q maps Xʹʹ Yʹʹ Zʹʹ back to XYZ
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