Tiêu đề matrices and determinnts1.pdf ✅

DEPARTMENT OF TECHNICAL EDUCATION, BENGALORE GOVERNMENT POLYTECHNIC, RAICHUR 2020-21 ENGINEERING MATHEMATICS UNIT-I, MATRICES AND DETERMINANTS STUDY MATERIAL PREPARED BY: RAMACHANDRA SUTAR N E A R G O V T I T I C O L L E G E , B I J A N G E R A R O A D A M A R K H E D L A Y P U T R A I C H U R - 584103
ENGINEERING MATHEMATICS UNIT-I, MATRICES AND DETERMINANTS GOVERNMENT POLYTECHNIC RAICHUR-584103 2 UNIT-I: MATRICES AND DETERMINANTS MATRICES 1.1 Introduction ➢ Matrices are most powerful tools in mathematics which simplifies many complicated methods into simple one. ➢ Matrices are originated with evaluation of solution of simultaneous linear equation. ➢ In electronics, mesh and nodal analysis lead to system of linear equation that can be described with a matrix. ➢ In computer science, matrices are used to project a three dimensional image to a two dimensional screen i.e, in graphics 1.2 Definition of matrix: The arrangement of numbers (or functions) in rows and columns wise enclosed within the brackets is called a matrix Examples: etc. 1.3 Order of a matrix: A matrix having m rows and n columns is called matrix of order mxn and read it as m by n or m cross n. Examples: etc. 1.4 Types of matrices: a. Square matrix: A matrix is said to be square matrix if its rows are equal to columns. Ex: b. Principal diagonal and Principal diagonal elements: In square matrix the diagonal from left top corner to right bottom corner is called principal diagonal or leading diagonal or primary diagonal and the elements lies on it are called principal diagonal elements. Ex: Here principal diagonal elements: 1, 5 and 9 c. Diagonal matrix: A square matrix is said to be diagonal matrix if all the elements are zero except principal diagonal elements Ex: These are the diagonal matrices d. Scalar matrix: A diagonal matrix is said to be scalar matrix if it’s all diagonal elements are equal. Ex these are Scalar matrices
ENGINEERING MATHEMATICS UNIT-I, MATRICES AND DETERMINANTS GOVERNMENT POLYTECHNIC RAICHUR-584103 3 e. Identity matrix or Unit matrix: A diagonal matrix is said to be Identity matrix if its all diagonal elements are equal to one.Usaualy identity matrices are denoted by “I” EX: these are Identity matrices. f. Symmetric matrix: A square matrix is said to be symmetric matrix if its all rows are interchanged into columns or columns into rows resultant is unaltered and fold the matrix on principal diagonal elements the overlapping elements are same. Ex: g. Skew-Symmetric matrix: A square matrix is said to Skew-Symmetric matrix if its all principal diagonal elements are zero and fold the matrix on principal diagonal elements the overlapping elements are same with opposite signs. Ex: h. Rectangular matrix: A matrix is said to be Rectangular matrix if its rows are not equal to columns. Ex:A = [ 1 2 ], B = [a b c], C = [ 2 0 3 7 6 5 ],D = [ a d b e c f ] and E = [ 8 9 6 ] etc i. Row matrix: A rectangular matrix is said to be Row matrix if it contains only one row. Ex:A = [3 5] and B = [x y z] etc are row matrices j. Column matrix; A rectangular matrix is said to column matrix if it contains only one column. Ex:A = [ 5 4 ] and B = [ p q r ] etc are column matrices k. Zero matrix or Null matrix; A matrix is said to zero matrix or null matrix if it all elements are zero. Usually zero matrices is denoted by 0. Ex:[ 0 0 0 0 ],[ 0 0 0 0 0 0 ],[0 0 0] etc are zero matrices. l. Upper triangular matrix: A matrix is said to be upper triangular matrix if all the elements below the principal diagonal elements are zero. Ex: A = [ −2 5 0 4 ] B = [ 2 3 −5 0 1 2 0 0 7 ] etc are upper triangular matrices. m. Lower triangular matrix: A matrix is said to be lower triangular matrix if it all the elements above the principal diagonal elements are zero. Ex: A = [ 4 0 3 2 ] B = [ 2 0 0 −5 3 0 −3 1 4 ] etc are lower triangular matrices. 1.5 Mathematical operations on matrices:
ENGINEERING MATHEMATICS UNIT-I, MATRICES AND DETERMINANTS GOVERNMENT POLYTECHNIC RAICHUR-584103 4 a. Equality of matrices: Two matrices A and B are said to be equal if they are of same order and each element of matrix A is equal to the corresponding element of matrix B. Ex:A matrices A = [ a d b c ] and B = [ p s q r ] are said to be equal if a = p, b = q, c = r and d = s. b. Transpose of matrix: The Transpose of a matrix A is denoted by A I or A T and is obtained by interchanging rows in to columns of matrix A. Ex: If A = [ 1 6 2 5 3 4 ] then A ′ = [ 1 2 3 6 5 4 ] Note: If A and B are matrices of suitable order and k is any scalar quantity then 1. (A ′ )′ = A 2. (kA)′ = kA′ 3. (A + B)′ = A′ + B′ 4. (AB)′ = B′A′ c. Addition of matrices: If A and B are two matrices of same order mXn then their addition A+B is also a matrix of order mXn and is obtained by adding corresponding elements of A and B. Ex: 1.If A = [ 1 4 2 3 ] and B = [ 2 3 5 2 ] then A + B = [ 1 + 2 4 + 3 2 + 5 3 + 2 ] = [ 3 7 7 5 ] 2.IF A = [ 1 −5 5 1 6 4 ] and B = [ 2 6 3 9 5 8 ] then A + B = [ 1 + 2 −5 + 6 5 + 3 1 + 9 6 + 5 4 + 8 ] = [ 3 1 8 10 11 12 ] Note: 1. The sum of two matrices A &B is obtained if and only if A and B are of same order 2. A+B=B+A [ Addition matrices is commutative] 3. A+(B+C)=(A+B)+C [Addition of matrices is associative] 4. A+0=0+A=A [ Identity exit with respect addition of matrices] 5. A+(-A)=(-A)+A=0 [ Inverse element exists with respect to addition] d. Subtraction of matrices: If A and B are matrices of same order mXn then their difference A-B is also a matrix of order mXn and is obtained by subtracting the elements of B from the corresponding elements of A Ex:If A = [ 8 1 9 4 ] and B = [ 2 −3 3 −2 ] then A − B = [ 8 − 2 1 + 3 9 − 3 4 + 2 ] = [ 6 4 6 6 ] Note: 1. The difference of two matrices A &B is obtained if and only if A and B are of same order e. Multiplication of matrices: The product AB of two matrices A and B is defined only if the number of columns in matrix A is equal to the number of rows in matrix B. That is if A is matrix of order mXn and B is matrix of order nXp then the product AB is a matrix of order mXp. Ex:If A [ 2 5 1 3 ] and B = [ 1 3 −2 4 ] then AB =? AB = [ 2 5 1 3 ] [ 1 3 −2 4 ] = [ 2X1 + 5X − 2 2X3 + 5X4 1X1 + 3X − 2 1X3 + 3X4 ] = [ 2 − 10 6 + 20 1 − 6 3 + 12] = [ −8 26 −5 15] f. Scalar multiplication of a matrix: If A is a matrix and k is a scalar then kA is a matrix obtained by multiplying each elements of A by the scalar k. Ex: 1.If A = [ a b c d ] then kA = [ ka kb kc kd]