UNIT 3:ACQUAINTED WITH TEACHING AND LEARNING
Key unit competence: Solve problem involving the system of linear equationsusing matrices.
3.0 INTRODUCTORY ACTIVITY
A Farmer Kalisa bought in Ruhango Market 5 Cocks and 4 Rabbits and
he paid 35,000Frw, on the following day, he bought in the same Market 3
Cocks and 6 Rabbits and he paid 30,000Frw.
a)Arrange what Kalisa bought according to their types in a simple tableas follows
b)Discuss and explain in your own words how you can determine the cost
of 1 Cocks and 1 Rabbit.
3.0 INTRODUCTORY ACTIVITY
Matrices provide a means of storing large quantities of information in such a
way that each piece can be easily identified and manipulated. They facilitate the
solution of large systems of linear equations to be carried out in a logical and
formal way so that computer implementation follows naturally. Applications of
matrices extend over many areas of engineering including electrical networkanalysis and robotics.
3.1. Definition and order of matrix
ACTIVITY 3.1
1) One shop sold 20 cell phones and 31 computers in a particular
month. Another shop sold 45 cell phones and 23 computers in
the same month. Present this information as an array of rows and columns.
2) a) Observe and complete the number of students in the year twoclasses on one Monday.
b) If every class gets new students on Tuesday such that in SME they
have 2 boys and 1 girls, in SSE they receive 1 girl and 1 boy, Complete
the table for new students.c) Complete the table for all students in an array of rows and columns.
CONTENT SUMMARY
A matrix is a rectangular arrangement of numbers or algebraic expressions which
illustrate the data for a real life model in rows and columns. A matrix is denoted
with a capital letter: A,B,C,…and the elements are enclosed by parenthesisor square brackets[ ].
Types of matrices
There are several types of matrices, but the most commonly used are1) Row matrix: matrix formed by one row
3.2. Operations on matrices
3.2.1 Addition and subtraction of matrices
ACTIVITY 3.2.1
1) In a survey of 900 people, the following information was obtained:
200 males thought federal defense spending was too high,150 males
thought federal defense spending was too low, 45 males had no
opinion, 315 females thought federal defense spending was too high
125 females thought federal defense spending was too low, 65
females had no opinion.Discuss and arrange these data in a rectangular array as follows:
3.3.2 Multiplying matrices
ACTIVITY 3.2.2
1) A clothing store sells men’s shirts for $40, silk ties for $20, and
wool suits for $400.
Last month, the store had sales consisting of 100 shirts, 200 ties, and
50 suits.
Using matrix, discuss and explain in your own words how todetermine the total revenue due to these sales.
CONTENT SUMMARY
Let A, B,C be matrices of order two or three
1) AssociativeA×(B×C) = ( A× B)×C
2) Multiplicative Identity
A× I = A, where I is the identity matrix with the same order as matrix A.
3) Not Commutative
A× B ≠ B× A
4) DistributiveA ×(B +C) = ( A× B) + ( A×C)
Find
a) The product A× B
b) The product B× Ac) Conclude about the commutativity of multiplication of matrices
Observation: The given matrices commute in multiplication.
Notice
• If AB = 0, it does not necessarily follow that A = 0 or B = 0 .
• Commuting matrices in multiplication:In general the multiplication of
matrices is not commutative, i.e, AB ≠ BA , but we can have the case where
two matrices A and B satisfy AB = BA. In this case A and B are said to be
commuting.
Trace of matrix
The sum of the entries on the leading diagonal of a square matrix, A, is knownas the trace of that matrix, notedtr ( A) .
3.4. Determinants and inverse of a matrix of order two and three
3.4.1. Determinant of order two or three
ACTIVITY 3.4.1
CONTENT SUMMARY
Consider two matrices, one of order two and another one of order three:
The determinant of A is calculated by SARRUS rule:
The terms with a positive sign are formed by the elements of the principal
diagonal and those of the parallel diagonals with its corresponding opposite
vertex.
The terms with a negative sign are formed by the elements of the secondary
diagonal and those of the parallel diagonals with its corresponding oppositevertex.
3.5. Applications of matrices and determinants
3.5.1 Solving System of linear equations using inverse matrix
ACTIVITY 3.5.1
A Farmer Kalisa bought in Ruhango Market 5 Cocks and 4 Rabbits
and he paid 35,000Frw, on the following day, he bought in the same
Market 3 Cocks and 6 Rabbits and he paid 30,000Frw.
a) Considering x as the cost for one cock and y the cost of one
Rabbit, formulate equations that illustrate the activity of Kalisa;b) Make a matrix A indicating the number of cocks and rabbits
Notice
• If at least, one of bi is different of zero the system is said to be non-homogeneous
and if all bi are zero the system is said to be homogeneous.
• The set of values of x, y, z that satisfy all the equations of system (1) is
called solution of the system.
• For the homogeneous system, the solution x = y = z = 0 is called trivial
solution. Other solutions are non-trivial solutions.
Non- homogeneous system cannot have a trivial solution as at least one ofx, y, z is not zero.
Note: The method of GAUSS helps us to solve the above system where CRAMER’S
method cannot.
