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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