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EDO STATE UNIVERSITY UZAIRUE
FACULTY OF ENGINEERING
ENGINEERING MATHEMATICS V
(GEE 412)
MODULE 1.0 _ MATRICES
BY
ENGR. OSEGBOWA E. DOUGLAS
OTHER LECTURERS:
ENGR. DR. (MRS) OBASA V. D
3. COURSE OUTLINE
1. Numerical Systems and Errors
2. Matrices and Related Topics
3. Solution of Linear Equations
4. Numerical Solution of Equations
5. Interpolation
6. Numerical integration and differentiation
7. Functional Approximation: Least Squares
Techniques
8. Characteristics values and vectors
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4. COURSE OUTLINE FOR MY PART
1. Matrices and Related Topics
2. Interpolation
• Finite Differences
• Linear and Quadratic Interpolation
• Interpolation with central differences(stirling
interpolation formula)
• Interpolation of Non Equally Spaced
data(Lagrange Interpolation
3. Numerical integration and differentiation
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5. INTRODUCTION TO MATRICES
• A matrix (plural matrices) is a rectangular array of
numbers, symbols, or expressions, arranged in
rows and columns enclosed within a bracket.
• The numbers, symbols, or expressions in a given
matrix are called elements.
• The horizontal and vertical lines of entries in a
matrix are called rows and columns, respectively.
• The dimension of a matrix is the order of the matrix
which is defined by the number of rows(R) and
columns(C) i.e RXC that it contains.
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6. INTRODUCTION TO MATRICES CONTD’
• A matrix with m rows and n columns is called an
m × n matrix or m-by-n matrix.
• Matrices are usually denoted with capital letters
while its elements are denoted with small letters.
Example 1
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7. Example 2
Matrix A = (a b)
The matrix A shown above is a 1 x 2 matrix, it has
one(1) row and two(2) columns, with
elements(entries) a, and b respectively.
Example 3
Matrix D =
a b c
d e f
g h i
The matrix D shown above is a 3 x 3 matrix, it has
three(3) rows and three(3) columns, with
elements(entries) a, b c, d, e, f, g, h, and i,
respectively.
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8. Example 4
Matrix B =
1 4
5 0
The matrix B shown above is a 2 x 2 matrix, it has
two(2) rows and two(2) columns, with
elements(entries) 0, 1, 4, and 5 respectively.
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Elements in an Array
• In any given matrix, the elements in the matrix
occupy a position on a row and column as shown:
A =
𝑎11 𝑎12 𝑎13
𝑎21 𝑎22 𝑎23
𝑎31 𝑎32 𝑎33
• In matrix A above, a11 the first no(1) denote the row
while the second no(1) denote the column, i.e the
element is located in the first row and first column.
10. Types of Matrices
1. Null or Zero Matrix: This is a matrix which has all
its elements to be zero(0), and its can be of any
order.
𝐴 =
0 0 0
0 0 0
0 0 0
2. Row and Column Matrix:
A = 𝑎 𝑏 𝑐 , B =
2
0
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11. 3. Square Matrix
This is matrix in which the no of rows is equal to the
no of columns. Example is a 2 x 2 and 3 x 3 etc.
A =
1 4
5 0
and B =
1 0 4
3 5 −1
−2 2 0
4. Unit or Identity Matrix(I)
D =
1 0
0 1
and F =
1 0 0
0 1 0
0 0 1
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5. Triangular Matrix: A square matrix where all the
elements below the left-right diagonal are 0 is called
an upper triangular matrix. A square matrix where all
the elements above the left-right diagonal are 0 is
called a lower triangular matrix.
1 2 3
0 5 4
0 0 2
Upper triangular matrix
5 0 0
4 1 0
7 5 2
Lower triangular matrix
13. 6. Transpose of a matrix: The transpose of matrix A
is written as A’ or AT
𝐴 =
2 3 1
5 6 7
0 8 9
; AT =
2 5 0
3 6 8
1 7 9
7. Symmetric Matrix: A matrix whose transpose is
the same as the original matrix is called a symmetric
matrix. Only a square matrix can be a symmetric
matrix.
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14. 8. Antisymmetric Matrix or Skew symmetric: Is a
square matrix whose transpose is its negation.
9. Orthogonal Matrix: A square matrix is called an
orthogonal matrix if the product of the matrix and its
transpose gives an identity matrix.
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15. 10. Equal Matrix: Two matrices are said to be
equal when they are of the same order, and the
elements in the corresponding positions are equal.
11. Singular Matrix: Singular matrix is a matrix
whose determinant is zero(0).
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