Engineering mathematics applies mathematical theory to complex real-world engineering problems through practical engineering, scientific computing, and combining traditional boundaries. Matrices were first formulated in 1850 and organize numbers and variables in a rectangular structure. They are widely used across many fields including chemistry to balance chemical equations represented as matrices, electrical circuits using Kirchhoff's laws, computer graphics for transformations, graph theory, cryptography through encoding/decoding matrices, seismic surveys, robotics for programming movements, analyzing forces on bridges, and recording data and reports.

1. APPLICATION OF
MATHEMATICS

2. Engineering Mathematics is the
art of applying mathematics to
complex real-world problems.
It combines
mathematical theory,
practical engineering and
scientific computing to address
today's technological
challenges. It is a creative and
exciting discipline, spanning
traditional boundaries.
ENGINEERING MATHEMATICS?.........

3. MATRICES
Who invented matrices?
Although a simple form of matrices may have been used
by the Mayans (and maybe other cultures; see below), the
true mathematical use of a matrix was first formulated
around 1850, by English mathematician, poet, and
musician James Sylvester (1814–1897).
• APPLICATIONS
• Chemical equations
• Electrical properties of a circuit
• Graph theory
• Cryptography
• Seismic Surveys
• Robotics
• Engineering :forces in a bridge or
truss
• A matrix organizes a group of
numbers , variables with specific
rules or arithmetic.
• It is represented as rectangular
group of columns & rows.

4. ---IN CHEMISTRY(CHEMICAL EQUATIONS)
Example:
aFeCl2 + bNa3(PO4)---> cFe3(PO4)2 + dNaCl
Fe: (1×a) + (0×b) –(3×c) = 0×d
Cl: (2 × a)+(0×b )+(0 × c) = 1 × d
Na: (0 × a)+(3 × b)+(0× c) = 1 × d
(PO4): (0×a)+(1×b)–(2 × c) = 0 × d
• In chemistry we use matrix
methodology to balance the
chemical equations.
• Initially we transform the
chemical equations into the
matrices form.
• Then we apply Gauss Jordon
method to calculate the
chemical reactions

5. ELECTRICAL PROPERTIES IN CIRCUITS
• Example:
• Circuit
From Kirchhoff’s law
• Loop1 e1=R1i+R3(i1-i2)
• Loop2 e2=R2i2+R3(i2-i1)
• E1=i1(R1+R3)-i2(R3)
• E2=-i1R3+i2(R2+R3)
• Matrices applied to electrical
circuits to calculate the amount of
current or voltage, resistance in a
given loop circuit.
• By applying Kirchhoff’s law the
equation is converted into
equations later we determine
the amount of voltage.
• Using Ohm’s law(v=i/R) from
voltage(v) we calculate the
resistance(R) and current(i)

6. Computer science
• Matrix transformers are useful within the world of computer graphics.
software and hardware graphics processors uses matrices for performing
operations such as scaling, translation , reflection and rotation
• Among the most common tools in electrical engineering and computer
science are rectangular grids of numbers known as matrices. The numbers
in a matrix can represent data, and they can also represent mathematical
equations.

7. Graph theory in matrices
Who invented graph theory?
This problem was first solved by the prolific Swiss mathematician Leonhard Euler, who, as a
consequence of his solution invented the branch of mathematics now known as graph theory.
• The adjacency matrix of a finite
graph is a basic notion of graph
theory. Linear combinations of
quantum states in Physics The first
model of quantum mechanics by
Heisenberg in 1925 represented the
theory's operators by infinite-
dimensional matrices acting on
quantum states. This is also referred
to as matrix mechanics . Computer
graphics 4×4 transformation rotation
matrices are commonly used in
computer graphics. Solving linear
equations Using Row reduction
Cramer's Rule ( Determinants) Using
the inverse Cryptography

8. Cryptography
• Application of matrix to Cryptography One type of code, which is extremely difficult to
break, makes use of a large matrix to encode a message. The receiver of the message
decodes it using the inverse of the matrix. This first matrix, used by the sender is called
the encoding matrix and its inverse is called the decoding matrix, which is used by the
receiver.
• Message to be sent: PREPARE TO NEGOTIATE and the encoding matrix be We assign a
number for each letter of the alphabet. Such that A is 1, B is 2, and so on. Also, we assign
the number 27 to space between two words.

9. Seismic surveys
• Many geologists make use certain
types of matrices for seismic
surveys. The seismic survey is one
form of geo physical survey that
aims at measuring the earth’s(geo)
properties by means of physical
principles such as magnetic, electric
, gravitational , thermal, and elastic
theories.

10. • In robotics and automation,
matrices are the base
elements for the robot
movements.
• The movements of the
robots are programmed in
MATLAB using matrix
pencil method.
• Matrices are used for
movements in Robot Arms.
ROBOTICS

11. Truss Bridges
Trusses are static beams which support bridges. For
the bridge to remain intact during use (one of the
important criteria of bridge design), it must be able
to withstand the force of objects traveling over it.
Linear algebra can be used to find how the forces
acting on the bridge will be distributed in the
trusses. Once the force on each truss is known, an
engineer can determine if the material of the truss
itself can withstand that force.

12. Other uses
• Matrices are useful for organizations , like for scientists who have to
record the data from their experiments if it includes numbers.
• In engineering , math reports are recorded using matrices.