EQT 101/3

Engineering
Mathematics I
By: Dr Nur Farhana Diyana Mohd Yunos
Faculty of Chemical Engineering Technology
Email: farhanadiyana@unimap.edu.my
CHAPTER 2: 01 Introduction
02 Types of Matrices
MATRICES
03 Operations
04 Determinants
05 Inverse
06 Systems of Linear Equations
 INTRODUCTION
A matrix is a rectangular array of elements or entries aij involving m rows and n
columns Columns, n

Rows, m • 2 matrices and

are said to be equal if m = r and n = s
then A = B.
• If aij for i = j, then the entries
a11,a22,a33,… are called the diagonal of
matrix A
Types of Matrices
Square Matrix Scalar Matrix
A diagonal matrix in which the diagonal
Matrix with order n x n elements are equal, aii = k and aij = 0 for i ≠ j
where k is a scalar

Diagonal Matrix
Matrix with order n x n with aij ≠ 0 and
aij = 0 for i ≠ j Identity Matrix
A diagonal matrix in which the diagonal elements
are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j
 Types of Matrices Upper Triangular Matrix
Zero Matrix If every elements below the diagonal is zero
A matrix which contains only zero or aij = 0, i > j
elements, aij = 0

DIAGONAL
Negative Matrix
A negative matrix of A =[aij] denoted by
–A where -A =[-aij] Lower Triangular
Matrix
If every elements above
the diagonal is zero or
aij = 0, i < j DIAGONAL
Types of Matrices
Transpose of Matrix
If A =[aij] is an m x n matrix, then the transpose
of A, AT =[aij]T is the n x m matrix defined by Properties Transposition Operation
[aij] = [aji]T
Let A and B matrices and k, .
Then,
 Types of Matrices
Symmetric Matrix Skew Symmetric Matrix
If AT = A, where the elements obey the If AT = - A, where the elements obey the rule
rule aij = aji aij = - aji, so that the diagonal must contain
zeroes.
Example 2
 Types of Matrices
LEADING 1
Row Echelon Form (REF)
Matrix A is said to be in REF if it
satisfies the following properties:
• Rows consisting entirely ZEROS ROW AT THE
zeroes occur at the bottom BOTTOM
of the matrix.
• For each row that doesn’t
consist entirely of zeroes, Name Here
LEADING 1
the 1st nonzero is 1.
• For each non zero row,
number 1 appear to the right
of the leading 1 of the
previous row.
LEADING 1
Types of Matrices
Reduced Row Echelon Form (RREF)
Matrix A is said to be in RREF if it
ZEROS ROW AT THE
satisfies the following properties:
BOTTOM • Rows consisting entirely zeroes
occur at the bottom of the matrix.
• For each row that doesn’t consist
entirely of zeroes, the 1st nonzero
LEADING 1 is 1.
• For each non zero row, number 1
appear to the right of the leading
1 of the previous row.
• If a column contains a leading 1,
then all other entries in the
column are zero
OPERATIONS OF MATRICES
Definition 2.15
Let and are matrices of order m x n. Matrices C=A+B is defined
by which is and Two matrices A and
B will be said conformable for addition only if they are both of the same order.
Definition 2.16
Let and are matrices of order m x n. . Matrices C=A-B is defined by
which is and Two matrices A and B will
be said conformable for subtraction only if they are both of the same order.

Properties of Addition and Subtraction

If then
OPERATIONS OF MATRICES
Definition 2.17
Let is an m xn matrix, , then the scalar multiplication is denoted

where
Properties of Scalar Multiplication

If and then

Definition 2.18
Suppose A is an m x n matrix
and B is a p x q matrix. For the
product AB to exist, it must be
that n=p, that is the number of
columns in A must be the
same as the number of rows in
B.
Example 3
Determinants: 2x2 Determinants: 3x3
Determinant of matrix A is defined by det(A) or |A|
Simple PowerPoint Presentation Definition 2.20
Definition 2.19
Simple PowerPoint Presentation
If Simple PowerPoint
is a 2x2 matrix, then the determinant Given is a 3x3
Presentation

Simple PowerPoint Presentation

is given by matrix, then the determinant is given
by:

Example
If and find det(A)
You can simply impress your audience and
add a unique zing and appeal to your
Presentations. Easy to change colors, photos
and det (B). Solution:
and Text. You can simply impress your
audience and add a unique zing and appeal to
your Presentations.
Determinants: 3x3 (minors &
cofactors)
Definition 2.21 Definition 2.22 Presentation
Simple PowerPoint

Simple PowerPoint
Cofactor of aij : Presentation
Let n ≥ 2 and A = [ aij ]nxn . Matrices
(n -1)x (n -1) submatrix of A is Simple PowerPoint Presentation

obtained by deleting the ith row and Simple PowerPoint Presentation

jth column of A, denoted by Mij . For 3x3 matrix :
Minor of
Therefore :

Cofactor of aij = You can simply impress your audience and

add a unique zing and appeal to your
We can conclude that : Even = 1, Odd = -1
Presentations. Easy to change colors, photos
and Text. You can simply impress your
audience and add a unique zing and appeal to
your Presentations.
Determinants: 3x3 (minors &
cofactors)
Simple PowerPoint Presentation Exercise 1:
Simple PowerPoint Presentation Find determinant of A and
Definition 2.23
Simple PowerPoint Presentation
If Cij is cofactor of matrix A, then cofactor A:
Simple PowerPoint Presentation
det(A) can be obtained by:
▪ Expanding along the ith row:

Cofactor
▪ Expanding along the jth column: =
You can simply impress your audience and
add a unique zing and appeal to your
Presentations. Easy to change colors, photos
and Text. You can simply impress your
determinant
audience and add a unique zing and appeal to
your Presentations.
of A:
Determinants: 3x3 (minors &
cofactors)
Exercise 2:
Find determinant of B and cofactor B:
Determinant Properties
Suppose A is n x n matrix and k is a Adding or subtraction a multiple of one
scalar. Suppose the matrix B is obtained row (column) to the other row (column)
by multiplying a single row or column of leaves the determinant unchanged.
A by k. Then det (B) = k det (A)

If matrix A is multiplied If A and B are 2 square

by k, that is every matrices such that AB
element in the matrix is exists, then, det(AB) =
multiplied by k, then det(A) det(B).
det(kA) = kn det(A)
If B is obtained from A by If 2 rows or 2 columns of
interchanging 2 rows or 2 a matrix are equal, the
columns, then determinant of the matrix
det(B) = - det(A) is zero.
 Solution
Adjoint
Cofactor =
Definition 2.24
Let A is an n x n matrix, then the
transpose of the matrix of cofactors A
is called the matrix adjoint to A.

Example
Find the adjoint of A:
Example
Find the adjoint of B:
Adjoint
Example
Find the adjoint of A:
 Inverse Definition
Inverse: 2 x 2
Definition 2.25 If , then is the inverse of A where
If A is a square matrix of order n and if
there exists a matrix A-1 such that
Theorem 1

then A-1 is called the inverse of A. If

Matrix A in invertible if and only if

If , then A doesn’t have an inverse.
Example
(i)Find the inverse for the given matrix:

Solution: Use definition

(ii) If , show that is the inverse of A.
if B is the
inverse of A
 Inverse: 3 x 3
1-Cofactor Method 2-Elementary
row Operations
(ERO)

Theorem 3
Let A and I both be n x n matrices, the
augmented matrix
Theorem 2
If A is nxn matrix, |A|≠0, then A-1 is
defined by:
may be reduced to by using
elementary row operation (ERO)
Inverse using the Cofactor Method
Find the inverse of each matrix using the
Cofactor Method: Step 4: find the inverse of A

Solution (a) Step 2: find det(A)

Step 1: find cofactor of A Step 3: find

adj(A)
Inverse using the Cofactor Method
Find the inverse of each matrix using the
Cofactor Method:
Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(i) : interchange the elements between ith row
and jth row

Example

ii) : multiply ith row by a nonzero

scalar, k
NEW R1
Example
Inverse using Elementary Row Operations (ERO)
Characteristics of ERO
(iii) : add or subtract ith row to a constant
multiple jth row by a nonzero
scalar, k
NEW R1
Example

Method of solving using ERO

Step 1: write A in augmented form

Step 2: use characteristics i,ii or iii to reduce A

Example
Find the inverse of each matrix using the
Elementary Row Operations (ERO)
 TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
A SYSTEM WITH NO SOLUTION

A SYSTEM WITH INFINITELY MANY A SYSTEM WITH UNIQUE

SOLUTION SOLUTION
Consider the
Consider the
systems: TYPES OF systems:
SOLUTIONS
Augmented Consider the
matrix: systems: Augmented
matrix:
Augmented
matrix:
Thus, the system has a unique
solution, where
METHODS OF SOLVING SYSTEMS OF EQUATIONS

Cramer’s Rule
04
Gauss-Jordan
03 Elimination
Gauss Elimination
02
The Inverse of the
01 Coefficient Matrix
 METHODS OF SOLVING SYSTEMS OF
EQUATIONS
A) The Inverse of the Coefficient Matrix Example
Consider a system of equations written in Solve the system by using the inverse of
the form AX=B, where the coefficient matrix:

Solution: Answer
Write in the form AX=B :

The solution of X is given by:

Where:
 METHODS OF SOLVING SYSTEMS OF
EQUATIONS
A) The Inverse of the Coefficient Matrix
Find the inverse of A, where

Find the solution of X:

Therefore;
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
Example
A. The Inverse of the Coefficient A four-ounce serving of Campbell’s®
Matrix Chicken & Beans contains 5 grams of protein
Example and 21 grams of carbohydrates. A typical
Solve the system by using the inverse of slice of “lite” rye bread contains 4 grams of
the coefficient matrix: protein and 12 grams of carbohydrates. Sara
planning a meal of “beans-on-toast” and she
want it to supply 20 grams of protein and 80
grams of carbohydrates.
How should she prepare her meal?
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
B. Gauss Elimination Example
Consider a system of linear equations: Solve the system by using the Gauss
elimination method.

Answer
:
The system can be written in the Solution
augmented form [A|B]. Write in augmented form: X=[A|B]
The augmented form: X=[A|B]
Apply ERO, use
appropriate
characteristics:

By using ERO, A may be reduce in REF/ Upper

Triangular form. (Refer Types of Matrices)
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
B. Gauss Elimination Apply backward
substitution:

x1 x2 x3
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
Example
You manage an ice cream factory that makes three
B. Gauss Elimination flavors: Creamy Vanilla, Continental Mocha, and
Example Succulent Strawberry. Into each batch of Creamy
Solve the system by using the Gauss Vanilla go two eggs, one cup of milk, and two cups of
elimination method. cream. Into each batch of Continental Mocha go one
(i) egg, one cup of milk, and two cups of cream. Into
each batch of Succulent Strawberry go one egg, two
cups of milk, and one cup of cream. Your stocks of
eggs, milk, and cream vary from day to day. How
(ii) many batches of each flavor should you make in order
to use up all of your ingredients if you have 350 eggs,
350 cups of milk and 400 cups of cream?
100 Creamy Vanilla flavour
50 Continental Mocha flavour
100 Succulent Strawberry flavour
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
C. Gauss-Jordan Elimination Example
Consider a system of linear equations: Solve the system by using the
Gauss-Jordan elimination method

Solution Answer
Write in augmented form: X=[A|B] :
The system can be written in the
augmented form [A|B].
The augmented form: X=[A|B]

Apply ERO, use appropriate

characteristics:

By using ERO, A may be reduce in RREF/

Identity form. (Refer Types of Matrices)
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
C. Gauss-Jordan Elimination

Therefore:
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
Example
The Arctic Juice Company makes three juice blends:
C. Gauss-Jordan Elimination PineOrange, using 2 quarts of pineapple juice and 2
Example quarts of orange juice per gallon; PineKiwi, using 3
Solve the system by using the quarts of pineapple juice and 1 quart of kiwi juice per
Gauss-Jordan elimination method. gallon; and OrangeKiwi, using 3 quarts of orange
(i) juice and 1 quart of kiwi juice per gallon. The amount
of each kind of juice the company has on hand varies
from day to day. How many gallons of each blend can
it make on a day if they have 650 quarts of pineapple
(ii) juice, 800 quarts of orange juice and 350 quarts of
kiwi juice?

100 gallons of PineOrange

150 gallons of PineKiwi
200 gallions of OrangeKiwi
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
D. Cramer’s Rule Cramer’s rule for a 3x3 system:
Given the systems of linear equation Since
as:

Then,
where

With D is the determinant of A

(D=det(A)) and D ≠ 0
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
D. Cramer’s Rule
Example
Where,
Solve the system by using the
Cramer’s rule method

Step of solution
(i) Compute D, determinant of A and D ≠ 0. Answer
(ii) Compute Di , i=1,2,3,…n, where Di is obtained :
from D by replacing the ith column of D by B.
Solution
(iii) Solution are given by Write down A, B and X:

(iv) Cramer’s rule cannot be applies if D = 0. In

such a case, either a unique solution to the
system does not exist or there is no solution.
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
D. Cramer’s Rule
Find D (determinant of A) Therefore:
 METHODS OF SOLVING SYSTEMS OF EQUATIONS
Example
The Arctic Juice Company makes three juice blends:
D. Cramer’s Rule
PineOrange, using 2 quarts of pineapple juice and 2
Example
quarts of orange juice per gallon; PineKiwi, using 3
Solve the system by using the
quarts of pineapple juice and 1 quart of kiwi juice per
Crames’r rule method:
gallon; and OrangeKiwi, using 3 quarts of orange
(i.)
juice and 1 quart of kiwi juice per gallon. The amount
of each kind of juice the company has on hand varies
from day to day. How many gallons of each blend can
it make on a day if they have 800 quarts of pineapple
(ii.)
juice, 650 quarts of orange juice and 350 quarts of
kiwi juice?

1 100 gallons of PineOrange

200 gallons of PineKiwi
150 gallions of OrangeKiwi
EIGENVALUES & EIGENVECTORS
 EIGENVALUES & EIGENVECTORS
Example 1 Solution
Consider matrix Since

Find the eigenvalues and eigenvectors using Hence, we have three possible eigenvalues.
Gaussian elimination method.
Step 2: Find eigenvectors corresponding to each
Solution: eigenvalue:
Therefore, When

Apply Gauss elimination to find eigenvectors, X.

 EIGENVALUES & EIGENVECTORS
Apply Gauss elimination on

Augmented matrix,
Solve from 2nd row:
Let and
From the 1st row:

Apply appropriate characteristics of ERO: Therefore, corresponding eigenvector for

EIGENVALUES & EIGENVECTORS
EIGENVALUES & EIGENVECTORS

Apply Gauss elimination to find eigenvectors, X.

Apply Gauss elimination on

Augmented matrix
EIGENVALUES & EIGENVECTORS
 EIGENVALUES & EIGENVECTORS
Example 2 Therefore,
Consider matrix

Find the eigenvalues and eigenvectors using

Gaussian elimination method.

Solution
Step 1: Find eigenvalue by computing

Sinc
e

Hence, we have three possible

 EIGENVALUES & EIGENVECTORS
Example 2 Augmented matrix

Apply appropriate characteristics of

ERO:

Solve from 1st

row:
 EIGENVALUES & EIGENVECTORS
Example 2 Apply Gauss elimination
Let, on
Therefore,
The corresponding eigenvectors to

Augmented matrix

Find eigenvectors corresponding to each

eigenvalue:
When

Apply Gauss elimination to find eigenvectors, X.

 EIGENVALUES & EIGENVECTORS
Example 2 Solve from 2nd
row:

Let,
Solve 1st row:

Therefore the corresponding eigenvector

for
Thank You