The presentation covers the definition and operations of matrices, including addition, subtraction, scalar multiplication, and multiplication, along with properties and applications in various fields such as geology, statistics, animation, and physics. Key concepts such as determinants, inverse matrices, and the representation of linear systems as matrix equations are also discussed. Overall, it highlights the significance of matrices in mathematical computations and real-world applications.

1. Presentation on
Matrix and it`sApplications
Presented by
Pritom Chaki
Roll: 19021060
MBA (Professional) Batch-21
Bangladesh University of Professionals
1

2. Contents
Definition of a Matrix
Operations of Matrices
Determinant of Matrix
Inverse of Matrix
Linear System to Matrix
Unique Properties of Matrix
Applications of Matrices
2

3. Matrix
(Basic
Definition)
Matrices are the rectangular agreement of
numbers, expressions, symbols which are
arranged in columns and rows.
A =
𝑎11 ⋯ 𝑎1𝑛
𝑎21 ⋯ 𝑎2𝑛
…
𝑎 𝑚1
…
…
…
𝑎 𝑚𝑛
= {𝐴𝑖𝑗}
3

4. Operations
with
Matrices
(Sum,
Difference)
If A and B have the same dimensions, then their
sum, A + B, is obtained by adding
corresponding entries.
In symbols, (𝑨 + 𝑩)𝒊𝒋 = 𝒂𝒊𝒋 + 𝒃𝒊𝒋.
If A and B have the same dimensions, then their
difference, A - B, is obtained by subtracting
corresponding entries.
In symbols, (𝑨 − 𝑩)𝒊𝒋 = 𝒂𝒊𝒋 - 𝒃𝒊𝒋.
4

5. Operations
with
Matrices
(Sum,
Difference)
Sum:
3 4 1
6 7 0
+
−1 0 7
6 5 1
=
2 4 8
12 12 1
The matrix 0 whose entries are all zero. Then, for all A , A + 0 =A
Difference:
2 4 8
12 12 1
-
3 4 1
6 7 0
=
−1 0 7
6 5 1
The matrix 0 whose entries are all zero. Then, for all A , A - 0 =A
5

6. Operations
with Matrices
(Scalar
Multiple)
If A is a matrix and r is a number (sometimes called a
scalar in this context), then the scalar multiple, rA, is
obtained by multiplying every entry in A by r.
In symbols, (𝑟𝐴)𝑖𝑗 = 𝑟𝑎𝑖𝑗 .
Example:
2
3 4 1
6 7 0
=
6 8 2
12 14 0
6

7. Operations
with Matrices
(Product)
If A has dimensions k × m and B has dimensions m × n,
then the product AB is defined, and has dimensions k × n.
The entry (𝑨𝑩)𝒊𝒋 is obtained by multiplying row i of A by
column j of B, which is done by multiplying corresponding
entries together and then adding the results i.e.,
Example:
(𝑎𝑖1 𝑎𝑖2 … 𝑎𝑖𝑚)
𝑏1𝑗
𝑏2𝑗
…
𝑏 𝑚𝑗
= 𝑎𝑖1 𝑏1𝑗 + 𝑎𝑖2 𝑏2𝑗 + ⋯ +𝑎𝑖𝑚 𝑏 𝑚𝑗
7

8. Operations
with
Matrices
(Product)
Example:
(𝒂𝒊𝟏 𝒂𝒊𝟐 … 𝒂𝒊𝒎)
𝒃 𝟏𝒋
𝒃 𝟐𝒋
…
𝒃 𝒎𝒋
= 𝒂𝒊𝟏 𝒃 𝟏𝒋 + 𝒂𝒊𝟐 𝒃 𝟐𝒋 + ⋯ +𝒂𝒊𝒎 𝒃 𝒎𝒋
𝒂 𝒃
𝒄 𝒅
𝒆 𝒇
𝑨 𝑩
𝑪 𝑫
=
𝒂𝑨 + 𝒃𝑪 𝒂𝑩 + 𝒃𝑫
𝒄𝑨 + 𝒅𝑪 𝒄𝑩 + 𝒃𝑫
𝒆𝑨 + 𝒇𝑪 𝒆𝑩 + 𝒇𝑫
8

9. Operations
with Matrices
(Transpose)
The transpose, 𝑨 𝑻 , of a matrix A is the matrix
obtained from A by writing its rows as columns.
If A is an k×n matrix and B = 𝑨 𝑻 then B is the n×k
matrix with 𝒃𝒊𝒋= 𝒂𝒋𝒊.
If 𝑨 𝑻=A, then A is symmetric
𝒂 𝟏𝟏 𝒂 𝟏𝟐 𝒂 𝟏𝟑
𝒂 𝟐𝟏 𝒂 𝟐𝟐 𝒂 𝟐𝟑
𝑻
=
𝒂 𝟏𝟏 𝒂 𝟐𝟏
𝒂 𝟏𝟐 𝒂 𝟐𝟐
𝒂 𝟏𝟑 𝒂 𝟐𝟑
9

10. Laws of
Matrix
Algebra
The matrix addition, subtraction, scalar
multiplication and matrix multiplication, have the
following properties.
Associative Laws:
A+ (B + C) = (A +B) + C, (AB)C = A(BC).
Commutative Law for Addition:
A + B = B + A
Distributive Laws:
A(B + C) = AB + AC, (A + B)C = AC + BC
10

11. Determinant
of Matrix
Determinant is a scalar
Defined for a square matrix
Is the sum of selected products of the elements of
the matrix each product being multiplied by +1 or -1
𝑑𝑒𝑡
𝑎 𝑏
𝑐 𝑑
= ad - bc
11

12. Inverse of
Matrix
Definition: An inverse matrix 𝑨−𝟏
which can be found
only for a square and a non-singular matrix A, is a
unique matrix satisfying the relationship
A 𝑨−𝟏
= 𝑰 = 𝑨−𝟏
A
The formula for deriving the inverse is
𝑨−𝟏
=
𝟏
𝒅𝒆𝒕(𝑨)
𝒂𝒅𝒋(𝑨)
12

13. System of
Equations in
Matrix Form
 The system of linear equations:
a11x1 + a12x2+a13x3+….+a1nxn = b1
a21x1 + a22x2+a23x3+….+a2nxn = b2
………………………………………
ak1x1 + ak2x2+ak3x3+….+aknxn = bk
Can be rewritten as the matrix equation Ax = b, where
A =
𝑎11 … 𝑎1𝑛
𝑎21
…
…
…
𝑎2𝑛
…
𝑎 𝑘1 … 𝑎 𝑘𝑛
, x =
𝑥1
𝑥2
…
𝑥 𝑛
, b =
𝑏1
𝑏2
…
𝑏 𝑘
13

14. Example:
Linear
System to
Matrix
Equation =
4𝑥 + 𝑦 + 2𝑧 = 4
5𝑥 + 2𝑦 + 𝑧 = 4
𝑥 + 3𝑧 = 3
A =
4 1 2
5 2 1
1 0 3
, x =
𝑥
𝑦
𝑧
, b =
4
4
3
AX = d
4 1 2
5 2 1
1 0 3
𝑥
𝑦
𝑧
=
4
4
3
14

15. Unique
Properties of
Matrices
In normal algebra , if we multiply two nonzero values,
then the outcome will never be a zero .
But if we multiply two non-zero values in matrix , then
the outcome can be zero.
Example: 𝑨 =
𝟑 𝟑
−𝟑 𝟑
and B=
𝟏 𝟏
𝟏 𝟏
AB =
𝟑 𝟑
−𝟑 𝟑
*
𝟏 𝟏
𝟏 𝟏
=
𝟑 ∗ 𝟏 + 𝟏 ∗ (−𝟑) 𝟑 ∗ 𝟏 + −𝟑 ∗ 𝟏
−𝟑 ∗ 𝟏 + 𝟑 ∗ 𝟏 −𝟑 ∗ 𝟏 + 𝟑 ∗ 𝟏
=
𝟑 − 𝟑 𝟑 − 𝟑
−𝟑 + 𝟑 −𝟑 + 𝟑
=
𝟎 𝟎
𝟎 𝟎
15

16. Application
of Matrix
Field of Geology
Taking Seismic Surveys
Plotting Graphs & Statistics
Scientific Analysis
Field of Statistics & Economics
Presenting real world data such as People's habit, traits
& survey data
Calculating GDP
16

17. Application
of Matrix
Field of Animation
Operating 3D software & Tools
Performing 3D scaling/Transforming
Giving reflection, rotation
Physics & Electronics
Elementary particles in quantum field theory
Traditional mesh & nodal analysis
Behavior of electronic components
17

18. 18
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