Chapter 6
Matrix Algebra
Chapter Objectives
• Concept of a matrix.
• Special types of matrices.
• Matrix addition and scalar multiplication operations.
• Express a system as a single matrix equation using matrix
multiplication.
• Matrix reduction to solve a linear system.
• Inverse matrix.
• Use a matrix to analyze the production of sectors of an economy.
• Matrices, the subject of this chapter, are simply arrays of numbers.
Matrices and matrix algebra have potential application whenever
numerical information can be meaningfully arranged into rectangular
blocks.
• One area of application for matrix algebra is computer graphics.
• An object in a coordinate system can be represented by a matrix that
contains the coordinates of each corner.
Matrix algebra
• Definition
Is a method to display and store the data
in the form of rectangular array and
organize the data.
Row & Column
Example
• A restaurant offers 4 kinds of pizza (vegetables, cheese, beef and
chicken) with 3 sizes.
Size
Kind
small
Medium
Large
Veg.
15
13
9
Chicken
16
18
12
Beef
13
10
8
cheese
18
20
7
• If we want to write itFirst
in column
a form of Second
matrix :
Third column
column
A=
Name of the
matrix
15
13
9
→ first row
16
18
12
→ second row
13
10
8
→ third row
18
20
7
→ fourth row
4 * 3 size of the matrix
Size of matrix = m x n , m → row
, n → column
Matrices
• A matrix consisting of m horizontal rows and n vertical columns is
called an m×n matrix or a matrix of size m×n.
 a11 a12
a
 21 a12
 .
.

.
 .
 .
.

am1 am 2
... a1n 
... a2n 
... . 

... . 
... . 

... amn 
For the entry aij, we call i the row subscript and j the column subscript.
Example 1 – Size of a Matrix
1
• a. The matrix
7 
• The matrix
1 3
has size
• b. The matrix
• The matrix
2 0
 1  6
5

1
 has size

9
4 
has size
1 3
.
32
.
1 1
7.
 2 4
9 11 5

6
8


6  2  1 has1 size1
35
.
Equality of Matrices
• Matrices A = [aij ] and B = [bij] are equal if they have the same size and
aij = bij for each i and j.
Example
Example
• Find x, y, and z
−1
0
𝑧 0 2
• If
=
4 2𝑦 − 3
4 7 5
• Solution
Z = -1
X + 5= 2 → x = 2 – 5 = -3
2y – 3 = 7 → 2 y = 7 + 3 → 2y = 10
→y=
10
2
=5
𝑥+5
5
Definition: Transpose of a matrix
• The Transpose of a Matrix A is denoted by AT
Example: Transpose of a Matrix
 1 2 3
A

4
5
6


• If
, find
• Solution
AT
.
 1 4
AT  2 5 
3 6 
Observe that
A 
T T
A
Example
2 −2
• If A = −3 4
−7 2
5
1 Find AT
5
• Solution
2 −3
• AT == −2 4
5
2
−7
2
5