Course: S5: Mathematics , Topic: Unit 1 Matrices and determinants

General

Unit 1 Matrices and determinants
Key Unit competence: Use matrices and determinants notations and properties to solve simple production, financial, economical, and mathematical related problems.
Introductory activity
The table below shows the revenue and expenses (in Rwandan francs) of a family over three consecutive months:

a) What was the family’s revenue in October?
b) By how much money did the family’s revenue increase from October to November?
1.1 Generalities on matrices
1.1.1. Definitions and notations
Learning Activity 1.1.1
A shop selling shirts records the number of each type of shirts it sells over a period of two weeks. In the first week, it sells 12 small size shirts, 8 medium shirts and 5 large shirts.
In the second week, it sells only 9 small size shirts and 3 medium size shirts.
a) What are the two criteria the shopkeeper will use to record these data?
b) Record this information in a rectangular array consisting of double entries.
c) Such a table is called a “matrix”. Describe the components of a matrix.

CONTENT SUMMARY
– A matrix is a rectangular arrangement of numbers, in rows and columns,
within brackets or [ ]. A matrix is denoted by a capital letter: A, B, C, ….
Rows are counted from the top of the matrix to the bottom of the matrix;
columns are counted from the leftmost side of the matrix to the rightmost side
of the matrix.
– The numbers in the matrix are called entries or elements.
The position of an entry in the matrix is shown by lower subscripts, such as ^aj :the entry on the i^th row and j^th column.
– If matrix A has n rows and p columns, then we say that the matrix A is of order n × p , read n by p, where the product n × p is the number of entries in the matrix.
Note: In finding the order of a matrix, we do not perform the multiplication n × p , we just write n × p , but for finding the number of entries of a matrix given by its order n × p ,we calculate the product n × p .
If A is a matrix of order n× p , then A can generally be written as A = (a_ij) ,where
i and j are positive integers, and; 1≤ i ≤ n ;1≤ j ≤ p .
– A matrix with only one row is said to be a row matrix; that is a matrix of order1× p .
Thus, (2 4 7) is a row matrix.
A matrix with only one column is said to be a column matrix; that is a matrix
of order n×1.
– A square matrix is a matrix in which the number of rows is equal to the number of columns; that is, matrix A of order n× p is a square matrix if and only if n = p ;
In this case, instead of saying a matrix of order n× n , we, sometimes, simply say a matrix of order n .
Application activity 1.1.1
1. Write down the order of each of the following matrices:

2. A shoe shop sells shoes for men and ladies. The first week, it sold 7
pairs of men’s shoes and 15 pairs of ladies’ shoes. The second week, it sold 9 pairs of ladies ‘shoes and 4 pairs of men’s shoes. Record this information as a 2× 2 matrix, stating what the rows stand for, and what the columns stand for
1.1.2. Equality of matrices
Learning Activity 1.1.2

Consider the following situations:
Situation1:
A class consists of boys and girls who are boarders or day scholars. The class teacher records the data by the matrix

and the rows represent the numbers of boarders and day scholars.
Situation2:
Two brothers sell shirts and shoes, in two different shop I and II, for two consecutive weeks. The Elder brother records his data by the matrix

shoes, and the rows represent the numbers of items sold in week1, and in
week2.
where the columns represent the numbers of shirts and shoes, and the rows
represent the numbers of items sold in week1, and in week2. Comment on the following, for matrices A, B and C:

a) Number of rows and columns
b) Corresponding entries (that is entries occupying the same
positions)
c) Nature of the elements.
d) Predict which two of the matrices above (A, B and C) are equal.
e) What are the conditions for two matrices to be equal?
CONTENT SUMMARY
Two matrices A = (a_ij)and B = (b_ij) are equal if and only if:
i) they have the same order;
ii) the corresponding entries (that is the entries occupying the same position,
in terms of rows and columns) are equal.
iii) The nature of the entries in the two matrices is the same.
Note: When discussing the equality of matrices, we assume the nature of the
entries in the two matrices to be the same.
A matrix, in which all the entries are zeros is said to be the null matrix or the
zero matrix.
For matrices, the equality A.B = 0 does not imply A = 0 or B = 0 , that is , the
product of two matrices can be the null matrix, yet none of the factors is a null
matrix.
1. 1. Determine whether the following matrices A and B are equal or not:

find the values of x, y, z and t .
1.2. Operations on matrices
1.2.1. Addition and subtraction of matrices
Learning Activity 1.2.1
A retailer sells two products, P and Q, in two shops, S and T.
She recorded the numbers of items sold for the last three weeks in each
shop by the following matrices:

a) Write down the order of each of the two matrices S and T. How are
these two orders?

b) Determine a single matrix for the total sales for this retailer for
the last three weeks in the two shops.

c) Predict the conditions for two matrices to be added and how to
obtain the sum of two matrices.

CONTENT SUMMARY
Matrices that have the same order can be added together, or subtracted. The
addition, or subtraction, is performed on each of the corresponding elements.

Application activity 1.2.1
1. Say, with reason, whether matrices A and B can be added or not. In
case they can be added, find their sum and the difference A− B
2. In a sector of a district, there are three secondary schools, A, B and C
having both boarding and day sections for both boys and girls. The
distribution of the students in the three schools are given, respectivel

where the first rows indicate the number of girls, the second rows
the number of boys, the first columns the number of boarders and
the second columns the number of day scholars in the three schools.

The Sector Education Officer (S.E.O) would like to record these data as a
single matrix S.
a) Which operation should he/she perform on the three matrices to
obtain matrix S?
b) Write down matrix S.
c) Use matrix S to answer the following questions:
i) How many day scholars are there from these three schools?
ii) How many girls are boarders from these three schools?
1.2.2. Scalar multiplication
Learning Activity 1.2.2
The monthly rental prices (in thousand Rwandan Francs) of three apartments without VAT (Value Added Tax)
are recorded by the matrix below:

M = (150 120 300) .
a) How do you calculate the VAT on an item?
b) What is the single operation to use in order to obtain the matrix M’
representing the monthly rental prices of the three apartments,
including 18% of VAT?
c) How do you obtain matrix M’?
d) Write down matrix M’

CONTENT SUMMARY
A matrix can be multiplied by a specific number; in this case, each entry of the matrix is multiplied by the given
number. This type of multiplication is called
scalar multiplication, since the matrix is multiplied by a single real number, and real numbers are also called scalars.

Example 1.2.2.

Solution:

Application activity 1.2.2

columns, and the numbers of shoes and clothes are in rows.
Since the festive period of Christmas is approaching, the shop
expects to double the number of each item to sell. Express the
resulting matrix D.
1.2.3. Multiplication of matrices
Learning Activity 1.2.3
Two friends Agnes(A) and Betty(B) can buy sugar, rice and beans at one

a) Compare the number of columns of M to the number of rows of P .
b) Calculate the shopping bill of each of the two friends at each of the
two supermarkets. Express the answer as matrix C .
c) How many rows and how many columns does C have?
d) Use matrix M and P to explain how each entry of C is obtained.

CONTENT SUMMARY
Let A and B be matrices of order n× p , and m× r , respectively. Matrices A and
B can be multiplied, in this order, if and only if p = m, that is, the number of
columns of the first matrix is equal to the number of rows of the second matrix.
In this case, we say that matrices A and B, in this order, are conformable for
multiplication. The product A× B is of order n× r , that is the product A× B
has the same number of rows as matrix A , and the same number of columns
as matrix B.
Practically, we proceed as follows:
1. Determine the order of the product:

Determine whether A and B , in any order, are conformable for
multiplication or not.
b) In case, they are conformable for multiplication, find the order of the
products A.B and B.A.What do you conclude about the multiplication
of matrices?
c) Find the matrix A.B
Solution:
a)
A : 2× 3
B : 3× 2
A.B : 2 × 2
A and B are conformable for multiplication, since the number of columns of A
is equal to the number of rows of B
In the same way,

B: 3× 2
A : 2× 3
B.A : 3 × 3
B and A are conformable for multiplication, since the number of columns of B
is equal to the number of rows of A.
b) The order of the product A.B is 2× 2 , and the order of the product
B.Ais 3×3 .
Multiplication of matrices is not commutative. In general, for matrices A and
B, A.B ≠ B.A

We can predict that multiplication of matrices is associative, that is, for all
matrices A, B and C , conformable for multiplication, (A.B).C = A.(B.C)

A matrix, in which all the entries are zeros is said to be the null matrix or the
zero matrix.
For matrices, the equality A.B = 0 does not imply A = 0 or B = 0 , that is , the
product of two matrices can be the null matrix, yet none of the factors is a null
matrix.
Application activity 1.2.3
1. Determine whether matrices A and B, in this order, are conformable
for multiplication or not. In case, they are conformable, find the
product:

represent the suppliers and the columns represent the prices.
Obtain the matrix for the total input bill for the next two months for both
suppliers.
1.2.4. Inversion of matrices
Learning Activity 1.2.4

CONTENT SUMMARY
– A square matrix with each element along the main diagonal (from
the top left to the bottom right) being equal to 1 and with all other
elements being 0 is said to be the identity matrix, it is denoted by I;
For any square matrix A of order n× n , and the identity matrix I of order n× n, we have:

A.I = A and I.A = A, that is, I is the identity element for multiplication of
matrices.
In particular,

– If for a square matrix A of order n × n , there exists a square matrix B of
order n × n , such that A.B = I and B.A = I , where I is the identity matrix of
order n× n , then B is said to be the inverse of matrix A , and written B = A⁻¹
– To find the inverse of a square matrix A ,of order n× n , by Gaussian method,
we, practically, proceed as follows:
Write ( A / I ) , a matrix of order n×(2n) , since the number of columns doubled,
but the number of rows is unchanged. Matrix ( A / I ) is an augmented matrix;
Transform the matrix ( A / I ) , using elementary row operations, to (I / B) .
Then B = A⁻¹ .
– The following are the elementary row operations:
1. Interchanging two rows. For example, if row 1 and row 2 are interchanged,
then the entries of row 1 become the respective entries of row2, and vice
versa; we write R₁ ↔ R₂
2. Multiplying each entry of a non-zero real number k .For example, if the
entries of row3 are multiplied by, say 2, we write R₃ → 2R₃
3. Adding to each entry of a row any multiple from any other row, for
example, R₁→ R₁ + kR₂
If matrices B exists, then we say that A is invertible or regular;
If B does not exist, then we say that A is a singular matrix.
Example 1.2.4.

Application activity 1.2.4

1.3. Determinants of square matrices
1.3.1.Definition and calculation of determinants of matrices of orders 2× 2 and 3×3

Learning Activity 1.3.1

CONTENT SUMMARY

Therefore, the determinant of a square matrix equals the sum of the products of
the entries on a row (or column) by their corresponding cofactors.
If the determinant of a square matrix is zero, then the matrix is singular; it has
no inverse.
If the determinant of a square matrix is not zero, then the matrix is invertible
or regular.
Example 1.3.1.

Solution:
The expansion along the second row will make the calculation of the
determinant easier.

Application activity 1.3.1

1.3.2. Properties of determinants
Learning Activity 1.3.2
1. Without calculation, predict the value of the determinant of each of
the following matrices:

CONTENT SUMMARY
A square matrix can be changed into simpler form
before calculating its determinant through properties including the following:
1. If all the entries of a row or column of a square matrix are zeros, then the
determinant of the matrix is zero.
2. If all the entries of a row (or column) of a square matrix are multiplied
by a real number k , then the determinant of the matrix is multiplied by

3. If two rows or columns of a square matrix are identical or proportional,
then the determinant of the matrix is zero.
4. If square matrix B is obtained by interchanging two rows or two columns
of square matrix A , then the determinant of B is the opposite of the
determinant of A .
5. 5)If a row or column of a square matrix B is obtained by adding or
subtracting any nonzero multiple of another row or column of matrix
A , the other rows or columns of B being the same as those of A ,then
the determinant of matrices A and B remains unchanged. Thus,

a) Column 2 and column 3 of matrix A are interchanged (C₂ ↔C₃ )

Application activity 1.3.2

1.4. Finding the inverse and solving simultaneous linear equations
1.4.1. Inverse of a matrix
Learning Activity 1.4.1

c) Perform the following:
i) Find det A
ii) Obtain matrix C ,where each entry of A is replaced by its cofactor
iii) Obtain matrix (denote it Adj(A) ) by writing the entries of the
first row of C as respective entries of the first column of Adj(A) ,
the entries on the second row of C as the respective entries of the
second column of Adj(A) , and the entries on the third row of C as
the respective entries of the third column of Adj(A)

CONTENT SUMMARY
Let A be a square matrix of order 2× 2 or 3×3 .Then the inverse of A . Can also
be calculated through the following four steps:
1. Find the determinant of A , that is det A;
2. Find the matrix C of cofactors of A : each entry of A is replaced by its cofactor.
3. Find the adjoint of matrix A , denoted, Adj(A) : the transpose of the matrix of cofactors;

Application activity 1.4.1

1.4.2. Solving simultaneous linear equations using inverse of a matrix
Learning Activity 1.4.2
A business makes floor tiles and wall tiles.
The table below shows the number of tiles of each type and the labor (in
hours) for making the tiles:

Given that the total cost for floor tiles is 53 (thousand) FRW and the total cost
for wall tiles is 37(thousand) FRW, find the material cost and the labor cost
by answering the following questions:
a) Label x the material cost and y the labor cost, and then model the
problem by simultaneous linear equations in x and y
b) Express the information in the table above as a matrix A of order
2× 2 , the total floor tile cost and the total wall tile cost as a matrix B
of order 2×1, and the material cost and the labor cost as a matrix X
of order 2×1
c) Perform the operation A.X = B and compare it to the simultaneous
equations obtained in part a)
d) Find the inverse matrix A⁻¹ and the product A⁻¹.B
e) Using X = A⁻¹.B ,find the values of x and y .
CONTENT SUMMARY
The two simultaneous linear equations in two unknowns, x and y ,

Therefore, the solution set of the simultaneous equations is S = {(2,3,−2)}
Application activity 1.4.2
Use matrices to solve the following simultaneous equations:

1.4.3. Solving simultaneous linear equations using Cramer’s rule
Learning Activity 1.4.3

1. a) Multiply both sides of equation(1) by b' to get equation(3), and
multiply both sides of equation (2) by −bto get equation(4)

b) Perform the addition (3) + (4) to obtain equation (5)
c) Make x the subject of formula in equation (5) , precising the
condition for this operation to be valid(possible).Label (6) this
equation.
d) Express the numerator and the denominator of (6) as determinants
of matrices of order 2× 2
2. a) Multiply both sides of equation(1) by −a ' to get equation (3') and
multiply both sides of equation (2) by a to get equation (4')
b) Perform the addition (3') + (4') to obtain equation (5')
c) Make x the subject of formula in equation (5') , precising the
condition for this operation to be valid(possible).Label (6') this
equation.
d) Express the numerator and the denominator of (6') as
determinants of matrices of order 2× 2
3. Use the formulas you have obtained above to solve the simultaneous
equations:

CONTENT SUMMARY
To solve the two simultaneous linear equation_s in t_wo unknowns ^x and ^y,
Cramer’s rule requires to go through the following steps:
In the same way, for the three simultaneous linear equations in three
Application activity 1.4.3
Use Cramer’s rule to solve the following simultaneous equations:

End of unit assessment 1
1. Write down the order of each of the following matrices:

2. Given that matrices A and B are equal, find the values of the letters:

3. Perform each of the following operations:

4. Invertible 2× 2matrices A, B and X are such that 4A− 5BX = B
a) Make X the subject of the formula
b) Find X if A = 2B

S5: Mathematics

General

Unit 1 Matrices and determinants

Unit 1 Matrices and determinants

Table of contents