Matrices and its applications

Basic concepts of matrices and its application

1. WELCOME

2. A. DALLY MARIA EVANGELINE ASSISTANT PROFESSOR PG & RESEARCH DEPARTMENT OF MATHEMATICS BON SECOURS COLLEGE FOR WOMEN THANJAVUR MATRICES

3. The term ‘matrix’ was first introduced by Sylvester in 1850. He defined a matrix to be an arrangement of terms. Knowledge of matrix is very useful and important as it has a wider application in almost every field of mathematics. The purpose of matrices is to provide a kind of mathematical shorthand to help the study of problems represented by the entries. The matrices may represent transformations of co-ordinate spaces or systems of simultaneous linear equations. INTRODUCTION

4. This representation gives the following informations. The elements along the rows represent the height, weight, age and pulse rate of the each patient. The elements along the first, second, third, fourth and fifth columns represent the height, weight, age and pulse rate of the patients. SIMPLE EXAMPLE

5. A matrix is a rectangular array or arrangement of entries or elements displayed in rows and columns put within a [] and (). The entries or elements may be any kind of numbers (real or complex), polynomials or other expressions. DEFINITION OF A MATRIX

6. The order or size of a matrix is the number of rows and the number of columns that are present in a matrix. Example of an m × n matrix ORDER OR SIZE OF A MATRIX

7. TYPES OF MATRICES

8. ROW AND COLUMN MATRICES A matrix having only one row - row matrix . Example: A = [1 -7 4] B = [22] C = [5 8 9 6] A matrix having only one column – column matrix Examples: A =  

9. SQUARE AND DIAGONAL MATRICES Matrix having equal number of rows and columns Order = n × n. Also called square matrix of order n Number of elements in a square matrix of order n is   Matrix having elements along the main diagonal Other entries are zero

10. SYMMETRIC MATRIX   SOME PROPERTIES OF SQUARE MATRIX

11. Let A be any square matrix. Then is symmetric A + B AB is symmetric iff AB = BA AB + BA is symmetric If A is symmetric, then kA is symmetric where k Є F   PROPERTIES OF SQUARE MATRICES

12. Addition and Subtraction Matrix multiplication Row-switching (swapping the rows) OPERATIONS OF MATRICES

13. Matrix addition is commutative A+B = B+A Matrix addition is associative A(B+C) = (A+B) +C Additive identity A + O = O + A = A Additive inverse A + (-A) = (-A) + A = O Matrix multiplication is not commutative Matrix multiplication is associative Matrix multiplication is distributive ALGEBRAIC PROPERTIES OF MATRICES

14. REAL LIFE APPLICATION OF MATRICES

15. MATRIX IN BIOLOGY EXTRACELLULAR MATRIX A meshwork of   proteins  and  carbohydrates   that binds cells together or divides one tissue from another.

16. MATRIX IN CHEMISTRY CRYSTAL DOT MODEL

17. 1 .  Use the  matrix  built-in function to create a data set based on the above function . 2. Plot the matrix and change the  Surface Fill  and the  Trace Color . Plotting a Matrix Created from a Function