MATRICES NOTE-2020

MATRICES [SWAIN’S MATH POINT] April 13, 2020 UNIT-II (ALGEBRA) CHAPTER-1 (MATRICES) __________________________________________________ 1.1 INTRODUCTION: Let the number of schools in different cities are given below. City School DAV SSVM KVS ST. XAVIER Berhampur 12 23 3 10 Sambalpur 21 34 43 5 Bhubaneswar 23 45 4 12 Rourkela 23 12 10 4 Then the total number of data can be expressed by a capital alphabet : ; ( , , , , , … ). 12 23 3 10 Now, = 21 34 43 5 23 45 4 12 23 12 10 4 Matrices: It is a rectangular array (arrangement) of numbers (real/complex) in a fixed order Order of Matrix:  The order of matrix is defined by × !\". $ = ⎡ ()) ()* ()+ ⋯ ()- ⎤ ⎢ (*) (** (*+ ⋯ (*- ⎥ ⎢ (+) (+* (++ (+- ⎥ ⎢ ⋯ ⎥ ⋮ ⋮ ⋮ ⋯⋯ ⋮ ⎢⋮ ⋮ ⋮ ⋮⎥ ⎣(/) (/* (/+ ⋯ (/-⎦3×4  Generally, it can be represented by ; $ = JKLMN3×4 Where, (56 is called the element of matrix with 78 row and 978 column.  3 × 4 is the order of matrix with 3 rows and 4 columns. Row: The horizontal lines formed by the elements are called rows. Column: The vertical lines formed by the elements are called columns. For example; (*+ means the element in a matrix with 2nd row and 3rd column. :; <=>×? ;> ; E?> <FG?×? @? AC @DC B < ;×? B ;×> E?H ; >>G?×; E?IG?×> :? ; >=>×; >;I F @H B FC D > ? ;×;





















MATRICES [SWAIN’S MATH POINT] April 13, 2020 1.5 SYMMETRIC & SKEW-SYMMETRIC MATRICES:  Symmetric Matrix: A square matrix is said to be symmetric matrix if. $ = $ˆ 1 −3 4 1 −3 4 Example: Let = @−3 5 6C Then, … = @−3 5 6C 4 68 4 68 Hence, = … Therefore, matrix is said to be symmetric matrix.  Skew-Symmetric Matrix: A square matrix is said to be symmetric matrix if, $ = −$ˆ 0 −5 4 0 5 −4 Example: Let = @ 5 0 −2C Then, … = @−5 0 2 C −4 2 0 4 −2 0 0 −5 4 Hence, = − … Now, − … = @ 5 0 −2C −4 2 0 Therefore, matrix is said to be Skew-symmetric matrix. NOTE: In Skew-symmetric matrix the diagonal elements must be ‘0’.  Theorem-1: For any matrix ,  + … is a symmetric matrix.  − … is a Skew-symmetric matrix. Proof: Let, = + … Now, to prove = … for symmetric matrix So, … = ( + …)… = … + ( …)… = …+ = +… = Therefore, = … (ho p e) Again, take = − … Now, to prove = − … for Skew-symmetric matrix So, … = ( − …)… = … − ( …)… = …− = −( − …) =− Therefore, = − … (ho p e)





















MATRICES [SWAIN’S MATH POINT] April 13, 2020 Problems: 1. Find the inverse of $ = E;? −I>G Solution: = E23 −41G | | = •23 −41• = 8 + 3 = 11 )) = 4 *) = 1 )* = −3 ** = 2 (e9 = E−43 21G ”) = > ((e9 ) |$| = > E−43 21G = @−43šš1111 12šš1111C >> >?? 2. Find the inverse of $ = @? > ?C ??> 122 Solution: = @2 1 2C 221 122 | | = ›2 1 2› = 1(1 − 4) − 2(2 − 4) + 2(4 − 2) = −3 + 4 + 4 = 5 221 )) = −3 *) = 2 +) = 2 )* = 2 ** = −3 +* = 2 )+ = 2 *+ = 2 ++ = −3 −3 2 2 (e9 = @ 2 −3 2 C 2 2 −3 ”) = > ((e9 ) |$| = > −3 2 2 < @2 −3 2C 2 −3 2 −; ? ? ⎡< < <⎤ ⎢ ?⎥ = ⎢ ? −; <⎥ ( \"Š) < < ⎢ ? ? −;⎥ ⎣< < <⎦