Presentation of Matrices IV

University of Babylon
College of Engineering
Department of Environmental Engineering
Engineering Analysis I (ENAN 103)







Matrices

Undergraduate Level, 3th Stage



Mr. Waleed Ali Tameemi
College of Engineering/ Babylon University
M.Sc. Civil Engineering/ the University of Kansas/ USA



2016-2017

Lecture Outline
Introduction
Matrix Types
Matrix Operations
The Determinant of a Matrix
The Adjugate of a Matrix
The Inverse of a Matrix














1.0 – Introduction
A matrix can be defined as an ordered rectangular array of numbers. Matrices are usually utilized in representing systems of linear equations, as will be explained in the system of linear algebraic equations lecture.
Matrix [A] contains n rows and m columns:
A_(n,m)=[?(?(a_11&a_12@a_21&a_22 )&?(?&a_1m@?&a_2m )@?(?&?@a_n1&a_n2 )&?(?&?@?&a_nm ))]

2.0 – Matrix Types
Row matrix
This type of matrix contains only one row and as follows:
R_(1,m)=[?(r_1&?(r_1&?&r_m ))]
Column Matrix
This type of matrix contains only one column and as follows:
C_(n,1)=[?(c_1@?(c_2@?@c_n ))]
Square Matrix
This type of matrix contains the same number of rows (n) and columns(n) and as follows:
S_(n,n)=[?(?(s_11&s_12@s_21&s_22 )&?(?&s_1n@?&s_2n )@?(?&?@s_n1&s_n2 )&?(?&?@?&s_nn ))]
Zero Matrix
A matrix with all zero entries:
Z_(n,m)=[?(0&0&0@0&0&0@0&0&0)]

Symmetric Matrix
In this type of matrix, if we change the row with the column, the matrix is still the same and as follows:
S_3,3=[?(1&2&3@2&9&4@3&4&6)]
Diagonal Matrix
All elements are zero except for the diagonal elements.
S_(n,n)=[?(?(s_11&0@0&s_22 )&?(?&0@?&0)@?(?&?@0&0)&?(?&?@?&s_nn ))]

Identical Matrix
It is a special type of diagonal matrix where all elements are zero except for the values of diagonal elements are equal to one.
S_(n,n)=[?(?(1&0@0&1)&?(?&0@?&0)@?(?&?@0&0)&?(?&?@?&1))]
Upper Triangular Matrix
In this type of matrix all elements below the main diagonal are equal to zero.
S_(n,n)=[?(?(s_11&s_12@0&s_22 )&?(?&s_1n@?&s_2n )@?(?&?@0&0)&?(?&?@?&s_nn ))]
Lowe Triangular Matrix
In this type of matrix all elements above the main diagonal are equal to zero.
A_(n,n)=[?(?(a_11&0@a_21&a_22 )&?(?&0@?&0)@?(?&?@a_n1&a_n2 )&?(?&?@?&a_nn ))]