MATRICES.pptx,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

n

1. MATRICES

2. Introduction Matrix Types of Matrices   Operations on Matrices  Transpose of a Matrix  Elementary Operation (Transformation) of a Matrix  Table of content

3. A matrix is a rectangular arrangement of a collection of numbers into a fixed number of rows and columns. When the elements are arranged in a matrix horizontally, it forms the rows of a matrix. When the elements are arranged in a matrix vertically, it forms the columns of a matrix . Matrices are usually denoted by capital letters of the alphabet . Order of a element   : m×n or n×n Introduction

4. Example When we refer to (point to) a certain elements of a Matrix, we usually use a symbol as follows. For a practice, let's suppose that we have a simple 3 x 3 matrix as shown below. (If you are not familiar with what 3 x 3 matrix  What is a(1,1) ? => It indicate Row 1 and Col 1 meaning '1' What is a(2,3) ? => It indicate Row 2 and Col 3 meaning '6' What is a(3,2) ? => It indicate Row 3 and Col 1 meaning '8'  

5. The range of operations that can be performed on a matrix involves three algebraic operations mainly. These three operations are namely -  Addition of matrices Subtraction of matrices Multiplication of matrices The basic operations among these are the addition and subtraction to matrices. For these operations, the matrices involved in the operation must be of identical order . However while multiplying, the number of columns of the first matrix must be equal to the number of rows of the second matrix. 

6. If A[ a ij ] mxn  and B[ b ij ] mxn  are two matrices of the same order, then their sum A + B is a matrix, and each element of that matrix is the sum of the corresponding elements, i.e. A + B = [ a ij  + b ij ] mxn Properties of Matrix Addition:  If A, B and C are matrices of the same order, then Addition of Matrices

7. We can subtract the matrices by subtracting each element of one matrix from the corresponding element of the second matrix, i.e. A – B = [ a ij  – b ij ] mxn . Subtraction of Matrices

8. If A and B be any two matrices, then their product AB will be defined only when the number of columns in A is equal to the number of rows in B. Properties of matrix multiplication Multiplication of matrices

9. The  transpose of a matrix  is obtained by changing its rows into columns (or equivalently, its columns into rows). A rectangular array of numbers or functions that are arranged in the form of rows and columns is called a  matrix . This array of numbers are called either entries or elements of a matrix. The transpose of the matrix is denoted by using the letter “T” in the superscript of the given matrix. For example, if “A” is the given matrix, then the transpose of the matrix is represented by A’ or A T . Transpose of a Matrix

10. Properties of tranaspose of a matrix

11. We know that  elementary row operations  are the operations that are performed on rows of a matrix. Similarly,  elementary column operations  are the operations that are performed on columns of a matrix. The three basic elementary operations or transformation of a matrix are: Interchange of any two rows or two columns. Multiplication of row or column by a non-zero number. Multiplication of row or column by a non-zero number and add the result to the other row or column. Elementary Operations of Matrix