Ch 4 - Matrix Algebra (2013) - Varsity · PDF fileChapter 4 Matrix Algebra ... Calculating the...

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Chapter 4 Matrix Algebra

4.2 Introduction to Matrix concepts

A matrix is a rectangular (or square) array or real numbers

arranged in rows and columns.

� = �� = �5 1 54 2 7�

The order (or dimension) of the matrix is a � × � where m

is the number of rows and n is the number of columns. A

matrix is square if � = �.

Vectors:

Row vector: Example: �1 2 4� is a 1 × 3 row vector.

Column vector: Example: �124� is a 3 × 1 column vector.

Scalar: Single number. Example: 4 is a scalar


4.3 Matrix operations

� = �1 −23 4 �, � = � 2 5−1 1�, � = �1 12 13 1�, � = �102� and

� = �−2 4�.

Matrix addition:

• � + � =

• � + � = � 2 5−1 1� + �102� ≠ Not possible to calculate


Matrix subtraction:

• � − � = �1 12 13 1� − �1 12 13 1� =

A matrix with all elements equal to zero is called a zero

matrix.

Transpose of a matrix:

• � = �−2 4� with �� = −2 (first row, first column)

and �� = 4 (first row, second column). �′ = �−24 �

• � = � 2 5−1 1� with �� = 2, �� = 5, �� = −1 and �� = 1. Therefore �′ =

• Note that the transpose of a row vector gives a column

vector & the transpose of a column vector gives a row

vector.


Multiplying a matrix by a scalar: 2� = 2�1 −23 4 � =

Matrix multiplication: �� = �1 −23 4 � � 2 5−1 1� = �1��2� + �−2��−1� = 4 �1��5� + �−2��1� = 3�3��2� + �4��−1� = 2 �3��5� + �4��1� = 19" =�4 32 19� .

�� = � 2 5−1 1� �1 −23 4 � =

Note that �� ≠ ��.

�� = � 2 5−1 1��1 12 13 1� ≠ Not possible to calculate.

The order of the matrix ��# is: $��×�� %��×�� $′��×�� = 1 × 1.


The Identity Matrix (I):

An Identity matrix is always square and has one’s on the

diagonal and zero’s elsewhere.

Let � = � 1 5−1 3� and � = �1 11 −22 0 �.

�& = � 1 5−1 3� �1 00 1� = � 1 5−1 3� = �.

&� = �1 00 1� � 1 5−1 3� = � 1 5−1 3� = �.

Note: In the previous ex. the Identity matrix is of order 2.

&� = �1 0 00 1 00 0 1��1 11 −22 0 � == �.

Note: In the previous ex. the Identity matrix is of order 3.


Take note: We DO NOT give you the Identity matrix. You

have to calculate the order yourself. Calculate: � + 4&


Practical example:

Model

L GL GLX GLE

Series A 500 300 280 90

Series B 400 150 220 190

Let ( = �500400300150280220 90190�, �� = �11� and �* = +1111,.

Then ��′ = �1 1� and �*′ = �1111�. Let .′ = �10131820� be the profit for the models.

Then . = /101318200.

Use matrix multiplication to calculate the following:

1. The total number of cars sold for series A and series B.


2. The total number of cars sold for each model.

3. Calculate the total profit of Series A and Series B.


4.5 The determinant of a matrix

• The determinant is indicated by |�|. • The matrix A must be square.

• |�| is a unique scalar value.

The determinant of a 2 × 2 matrix:

For a 2 × 2 matrix � = �� the determinant is

calculated using |�| = 3�� 3 = �� − ��.

Example � = � 2 5−1 1�.

Then |�| = 3 2 5−1 13 =

Property 1

The determinant of a matrix, A, has the same value as that of

its transpose matrix, A’. That is |�| = |�′|. If � = � 2 5

−1 1� then |�| = 7.

Now �′ = �2 −15 1 � so that 4�′4 = 32 −1

5 1 3 =


Property 2

The interchanging of any two rows (or two columns) will

alter the sign, but not the numerical value, of the

determinant.

If � = � 2 5−1 1� then |�| = 7.

Interchanging two rows: 3−1 12 53 =

If � = � 2 5−1 1� then |�| = 7. Interchanging two columns: 35 21 −13 =

Property 3

The multiplication of any one row (or column) by a scalar k,

will change the value of the determinant k-fold.

If � = � 2 5−1 1� then |�| = 7. Multiplying the first row by 5 = 10:

320 50−1 1 3 =


Property 4

If each element of a row (or column) is added to (or

subtracted from) the corresponding element of another row

(or column), the value of the determinant remains the same.

If � = � 2 5−1 1� then |�| = 7. Add the first row and the

second row and place the sum in the first row:

3 1 6−1 13 =

Property 5

If one row (or column) is a multiple of another row (or

column) then the determinant is zero. |�| = 31 42 83 =

Note: the second row is a multiple (× 2) of the first row.

Property 6

If one row (or column) of a matrix contains only zeros then

the value of the determinant is zero. |�| = 31 20 03 =


The determinant of a 7 × 7 matrix:

|�| = 81 0 22 3 03 4 −28 =

• If |�| = 0 then the matrix A is singular.

• If |�| ≠ 0 then the matrix A is non-singular.


Cramer’s rule:

Consider the set of two linear equations: ��9� + ��9� = �� 9� + ��9� = ��

Matrix notation: �: = � with � = �� , : = �9�9��

and � = ��".

The linear equations can be solved using Cramer’s rule

9� = ;<= >=?<? >??;|%| and 9� = ;>== <=>?= <?;|%| .

Note: This holds if A is non-singular (|�| ≠ 0�.


Example 29� − 59� = 31 89� + 99� = −21

Matrix notation: �: = � with � = �2 −58 9 �, : = �9�9�� and � = � 31−21�.

|�| = 32 −58 9 3 =

9� = 3 �� @A@�� B 3|%| =

and

9� = 3� ��C @��3|%| =


Consider the set of three linear equations: ��9� + ��9� + ��9� = �� 9� + ��9� + ��9� = �� 9� + ��9� + ��9� = ��

Matrix notation: �: = � with � = �� ,

: = �9�9�9�� and � = ��.

The linear equations can be solved using Cramer’s rule

9� = 8<= >=? >=D<? >?? >?D<D >D? >DD8|%|

and

9� = 8>== <= >=D>?= <? >?D>D= <D >DD8|%|

and

9� = 8>== >=? <=>?= >?? <?>D= >D? <D8|%| .


Example 9� − 29� = 1 49� − 29� + 9� = 2 9� + 29� − 109� = −1

Matrix notation: �: = � with � = �1 0 −24 −2 11 2 −10�,

: = �9�9�9�� and � = � 12−1�.

|�| = 81 0 −24 −2 11 2 −108 = −2

9� = 8 � E @�� @� �@� � @�E8|%| =

9� = 8� � @�* � �� @� @�E8|%| =

9� = 8� E �* @� �� @�8|%| =


The inverse of a matrix:

If there exists a square, non-singular matrix A then there

exists a square matrix �@� such that �@�� = & and ��@� = & then �@� is the inverse matrix of

A.

Let � = �3 54 7� and test if � 7 −5−4 3 � is the inverse of

matrix A.

��@� = �3 54 7� � 7 −5−4 3 � = �1 00 1�.

Note: You do not have to calculate the inverse of a matrix

by hand, but you have to be able to calculate the inverse of a

matrix using Microsoft Excel.


The matrix �@� is used to solve a set of linear equations

�: = � �@��:� = �@�� &: = �@�� : = �@��

Example 39� + 59� = 1 49� + 79� = 0

Solve 9� and 9� using the inverse matrix.

Given: �3 54 7�@� = � 7 −5−4 3 �

�: = � �3 54 7� �9�9�� = �10�

Therefore, : = �@�� 9�9�� = � 7 −5−4 3 � �10� =


Example 89� − 9� = 15 9� + 59� = 1 29� + 39� = 4

Solve 9�, 9� and 9� using the inverse matrix.

Given: �8 −1 00 1 52 0 3�@� =

FGH ��* ��* @A�*�E�* �*�* @*E�*@��* @��* C�* IJ

K =��*� 3 3 −510 24 −40−2 −2 8 �

�: = � �8 −1 00 1 52 0 3��9�9�9�� = �1514 �

Therefore, : = �@��

�9�9�9�� = ��*� 3 3 −510 24 −40−2 −2 8 ��1514 � =


Practical applications:

Example

Consider the sales of 3 shops at a motor show. Flags, T-

shirts and caps are sold.

Shop Flags T-shirts Caps

1 8 4 10

2 2 3 14

3 5 0 6

The sales matrix ( = �8 4 102 3 145 0 6 �.

Suppose the prices of the items are known and are the same

for the 3 shops. The prices are R10 for a flag, R8 for a T-

shirt and R5 for a cap. Let . = �1085 � and �� = �111�.

Use matrix multiplication to calculate the following:

1. The total number of items sold at the 3 shops.


2. The total number of flags, T-shirts and caps sold.

3. The total income for each shop.


Suppose the income for the 3 shops is known and the prices

of the items are the same at the 3 shops.

Shop Income

1 162

2 114

3 80

To calculate the prices for the items we need to solve for the

following linear equations: 89� + 49� + 109� = 162 29� + 39� + 149� = 114 59� + 09� + 69� = 80

or (: = � such that �8 4 102 3 145 0 6 ��9�9�9�� = �16211480 �. The following is given: |(| = 226 and (@� = �� 9 −12 1329 −1 −46−7.5 10 8 �.

Use the information and calculate the following:

1. Test if (@� is the inverse matrix of S.


2. Calculate the price of a flag by using Cramer’s rule.

3. Calculate the price of a flag by using the inverse matrix.


Excel functions for Matrix Algebra:

Calculating the transpose of a matrix using Excel.

Formula worksheet:

Press Ctrl + Shift + Enter

Value worksheet:


Calculating the determinant of a matrix using Excel.

Formula worksheet:

Press Enter

Value worksheet:


Calculating the inverse of a matrix using Excel:

Formula worksheet:


Value worksheet:


Multiplying two matrices using Excel:

Formula worksheet:


Value worksheet:

Ch 4 - Matrix Algebra (2013) - Varsity · PDF fileChapter 4 Matrix Algebra ... Calculating the...

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