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Good morning students, Once again we recall previous topic, i.e. matrices introduction The matrix was first introduced by a French mathematician cayley in the year 1857. The theory of matrices is one of the powerful tools of mathematics not only in the field of higher mathematics but also in other branches such as applied sciences, nuclear physics, etc. Definition of matrix:           A Matrix is a rectangular array of numbers (real or complex) arranged in to rows and columns enclosed by square brackets. Example1:  A=[■(1&2@3&4)] Example2: B=[■(1&2&3@4&5&6@8&9&10)] Usally the matrices are denoted by capital letters of English alphabets A,B,C…. and the elements of the matrices are represented by small letters. Note: In generally matrices are enclosed with square brackets Type equation here. A GENERAL REAL MATRIX (OR) STRUCTURE OF A MATRIX A=[■(a_11&〖        a〗_(12…     … …)&a_1n@〖a 〗_█(21@⋮@⋮@⋮)&         a_█(22 … ……@⋮@⋮@⋮)…&a_█(2n@⋮@⋮@⋮)@a_m1&a_m2&a_mn )]   this is a mxn matrix. Its written as mxn and read as m by n The horizontal lines are called rows and the vertical lines are called columns Note: The rows are counted from top to bottom and the columns are counted from left to right Order of a matrix: if there are m rows and n columns in a matrix, then the order of the matrix is mxn  or m by n. Example: A=[■(2&3@-1&5)] there are two rows and two columns hence it is a 2x2 (or) 2by 2 matrix. Types of matrices: Row matrix: A matrix having only one row is called a row matrix. Example: A=[■(1&■(2&3&4))] Column matrix: A matrix having only one column is called a column matrix.

Example: A= [■(1@2)] Square matrix: A matrix which has equal number of rows and columns is called a square matrix. Example: A=[■(2&3@-1&-6)] Example: B =[■(2&1&0@-5&6&7@1&0&8)] Rectangular matrix: A matrix which is not square then its called rectangular matrix. Example: A =[■(1&2&4@3&5&7)] It is a rectangular matrix. Diagonal matrix: A diagonal matrix is nothing but it’s a  square matrix, in this only diagonal elements are non-zero and all other elements are zero. Example: A=[■(2&0@0&-6)] Scalar matrix: it’s a type of diagonal matrix.in this all the diagonal elements are same.

Example: B =[■(2&0&0@0&2&0@0&0&2)]

Identity matrix: An identity matrix is a type of scalar matrix, in which all the diagonal elemtns are unit. it is denoted by I. not with any other alphabet. Example: A=[■(1&0@0&1)] it is a identity matrix of order 2 by 2. it is denoted by I_2 Example: B =[■(1&0&0@0&1&0@0&0&1)] it is a identity matrix of order 3 by 3.it is denoted by I_3. QUIZ: Multiple choice questions: A matrix is said to be a Square matrix if and only if                                  	order of 2x3 order of 1x2 order of mxm none of the above An identity matrix having Principal diagonal elements all are unit Principal diagonal elements all are equal to two None principal diagonal elements are unit None of the above The horizontal lines of the matrices are called Columns Rows Principal diagonal Matrix The vertical lines of the matrices are called Matrix Pricipal diagonal Scalar matrix Columns A scalar matrix Having principal diagonal elements are same and all other elements are zero Having principal diagonal elements are equal to unity None diagonal elements are zero All elements are equal to zero