GEE 412 _ Module 1.0 _ Matrices _ by Engr. Douglas.pptx
- 1. 10/29/2023 1 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
- 2. COURSE INFORMATION COURSE TITLE: Engineering Mathematics COURSE CODE: GEE 412 COURSE UNIT: 3 TERM: FIRST SEMESTER 2022/2023 MEETING TIMES: TUESDAY (1 – 3 am) 10/29/2023 2
- 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 10/29/2023 3
- 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 10/29/2023 4
- 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. 10/29/2023 5
- 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 10/29/2023 6
- 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. 10/29/2023 7
- 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. 10/29/2023 8
- 9. 10/29/2023 9 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 10/29/2023 10
- 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 10/29/2023 11
- 12. 10/29/2023 12 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. 10/29/2023 13
- 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. 10/29/2023 14
- 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). 10/29/2023 15
- 16. Assignments Briefly discuss four(4) other types of matrices with examples other than the just discussed types. 10/29/2023 16
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