Scene 1

MATRICES - By Group B9 Ketan   Tarwani , FYBFM101 Aryan  Tejwani , FYBFM102 Krishna  Tulsian , FYBFM103 Janvi   Vakil . FYBFM 104 Riyansh Vanigotta, FYBFM 105

Scene 2

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns .   For example, the dimension of the matrix below is 2 × 3 (read "two by three"), because there are two rows and three  columns:

Scene 3

The term  matrix  was introduced by the 19th-century English mathematician  James Sylvester It was his friend the mathematician Arthur  Cayley  who developed the algebraic aspect of matrices in two papers in the 1850s. Cayley   first applied them to the study of systems of linear equations, where they are still very  useful.  Matrices have wide applications in engineering, physics, economics, and statistics as well as in various branches of mathematics.

Scene 4

The number of rows and columns that a  matrix  has is called its  order  or its  dimension . By convention, rows are listed first; and columns, second . The Dimension of a matrix can help us determine the number of elements in a matrix. The dimension is obtained by multiplying the number of rows and  collumns .  Thus, we would say that the order (or dimension) of the matrix below is 3 x 4, meaning that it has 3 rows and 4 columns .

Scene 5

Matrices are often denoted using capital roman letters such as A, B and C . The  individual items in an  m × n  matrix  A , often denoted by  a i , j , where  i  and  j  usually vary from 1 to  m  and  n , respectively, are called its  elements  or  entries . Hence, The individual items in an  m × n  matrix  A , often denoted by  a i , j , where  i  and  j  usually vary from 1 to  m  and  n , respectively, are called its  elements  or  entries .

Scene 6

Let us take an example to understand the notations of matrices.  A=                                                                      B=  The  matrix is denoted by an upper case letter and its elements are denoted by the same letter in the lower case .  a ij    denotes  any element of matrix   A which  is  in  i th  row and  j th  column. Similarly  b ij   denotes any element of matrix B  So in the matrix A given above, the element a 21  denotes the element which is in the second row and the first column of Matrix A   Therefore,  a 21  = 12   Similarly, b 32  = 9 and b 23  = (-23)

Scene 7

Null Matrix or a Zero Matrix If  in a matrix all the elements are zero then it is called a  zero matrix  and it is generally denoted by 0 .   Thus, A = [ a ij ] mxn  is a zero-matrix if  a ij  = 0 for all i and  j For  eg , A is a null matrix of the order 3X3 and B is a null matrix of the order 2X5 A=                                                           B=

Scene 8

Row Matrix-  A matrix having only one row is called a row matrix. Thus A = [ aij ] mxn  is a row matrix if m = 1. So, a row matrix can be represented as A = [ aij ]1×n. It is called so because it has only one row and the order of a row matrix will hence be 1 × n.  For  example, A = [1 2 4 5] is row matrix of order 1 x 4. Another example of the row matrix is P = [ -4 -21 -17 ] which is of the order 1×3.  Collumn  Matrix-  A matrix having only one column is called a  column matrix . Thus, A = [ a ij ] mxn  is a column matrix if n = 1. Thus, the value of for a column matrix will be 1. Hence, the order is m × 1 . For example, B is a column matrix of the order  3X1  if  B=

Scene 9

Square Matrix-  If the number of rows and the number of columns in a matrix are equal, then it is called a  square matrix . C is a square matrix if it is denoted as C ij  where i=j For example, A is a square matrix of the order 3X3 and B is a square matrix of the order 2X2 IF  A=                                                                        B=

Scene 10

Diagonal Matrices-  If all the elements, except the principal diagonal, in a square matrix, are zero, it is called a  diagonal matrix . Thus, a square matrix A = [ a ij ] is a diagonal matrix if  a ij  = 0,when  I  /    j  For example, A is  s a diagonal matrix of order 3 x 3, which can also be denoted by diagonal [2 3 4]. The special thing is, all the non-diagonal elements of this matrix are zero . A= A diagonal matrix is always a square  matrix.

Scene 11

Scalar Matrix-  If all the elements in the diagonal of a diagonal matrix are equal, it is called a  scalar  matrix.  Thus A is a scalar matrix of the order 3X3 as all the elements in the diagonal of A are equal.  ( a 11  = a 22  = a 33  = 5) A= Identity or Unit Matrix-  If all the elements of a principal diagonal in a diagonal matrix are 1, then it is called a  unit matrix . A unit matrix of order n is denoted by  I n Thus B is a unit matrix of the order 2X2 as all the elements in the diagonal of A are B.  B=

Scene 12

Triangular Matrix-  A square matrix is said to be a  triangular matrix  if the elements above or below the principal diagonal are zero. There are two  types: Upper Triangular  MatriX -  A  square matrix [ a ij ] is called an  upper triangular matrix , if  a ij  = 0, when i > j . Thus A is an upper triangular matrix of the order 3X3as it satisfies the above criteria.  A=  Lower Triangular  Matrix - A  square matrix is called a  lower triangular matrix , if  a ij  = 0 when i < j. Thus  B is a lower  triangular  matrix of the order 3X3  as it satisfies the above  criteria. B=

Scene 13

The  transpose of a matrix  can be defined as an operator which can switch the rows and column indices of a matrix i.e. it flips a matrix over its diagonal . For example, A is a matrix of the order 4X3. It’s transpose matrix, denoted as A’ will be a matrix of the order 3X4 with the elements arranged in such a manner that       A ij    =  A’ ji A=                                                                                    B=

Scene 14

Symmetric Matrix and Skew-Symmetric Matrices-  A symmetric matrix and skew-symmetric matrix both are square matrices. But the difference between them is, the symmetric  matrix  is equal to its transpose whereas skew-symmetric matrix is a matrix whose transpose is equal to its negative. If A is a symmetric matrix, then A = A T   and if A is a skew-symmetric matrix then A T  = – A. For example, A is a symmetric matrix as A=A’. B is a skew-symmetric matric as B= -(B’) A=                              A’=                              B=                                 B’=

Scene 15

A ddition  of algebraic expressions, we know that while the addition of algebraic expressions can only be done with the corresponding like terms, similarly the addition of two matrices can be done by addition of corresponding terms in the matrix . There are basically two criteria which define the addition of matrix. They are as follows:  Consider two matrices A & B. These matrices can be added  if(if  and only if) the  order of the matrices  are equal, i.e. the two matrices have the same number of rows and columns. For example, say matrix A is of the order  3X4  then the matrix B can be added to matrix A if the order of B is  also 3X4 The addition of matrices is not defined for matrices of different sizes.

Scene 16

Example for Matrix Addition Matrices A and B are to be added. The order of both the matrices is 2X2. The order of the resultant matrix A+B is also 2X2.  A=                                                      B=  A+B= A+B=

Scene 17

Matrix subtraction is exactly the same as matrix addition. All the constraints valid for addition are also valid for matrix subtraction. Matrix subtraction can only be done when the two matrices are of the same size. Subtraction cannot be defined for matrices of different sizes. Mathematically , P – Q= P+ (-Q) Matrices A and B are to be  subtracted.  The order of both the matrices is 2X2. The order of the resultant matrix  A-B  is also 2X2 . A=                                                 B=       A-B=                               =

Scene 18

Scene 19

For example, Matrices A and B of the orders 2X3 and 3X2 can be multiplied as the number of rows in Matrix A are equal to number of columns in Matrix B are equal.  A=                                                               B= AB= AB=

(2X3) + (1X4)  + (2X0) (2X2) + (1X1) + (2X2) (1X3) + (3X4) + (4X0) (1X2) + (3X1) +  (4X2)

Scene 20