WUSTL ESE 318 - Lecture 5 - Matrix Intro(1) (8 pages)

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Lecture 5 - Matrix Intro(1)



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Lecture 5 - Matrix Intro(1)

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Pages:
8
School:
Washington University in St. Louis
Course:
Ese 318 - Engineering Mathematics A

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ESE 318 02 Fall 2014 Lecture 5 Matrix Intro Systems of Equations Sept 9 2014 Section 8 1 Matrix Review Algebra Yellow highlights were filled in by students during the class Definition A matrix is a rectangular array of numbers a11 a 21 A a m11 a12 a 22 am2 a1n a2n a mn The elements are designated by row and column The jk element is the number in row j and column k of the matrix Size We say it has m rows and n columns and its size is m X n pronounced m by n Some examples below 7 4 2 1 5 6 9 3 8 3 3 Size x y 2 1 a b 1 3 1 2 0 c 0 3 5 3 2 Addition and Subtraction Only matrices of the same size can be added Addition is by element as follows 2 3 4 1 0 0 7 1 6 5 4 2 1 6 7 0 3 1 2 2 0 3 2 4 0 3 1 5 4 7 1 0 0 3 0 1 7 2 1 2 6 0 5 3 6 2 0 4 6 5 11 1 3 3 1 9 8 2 6 11 Subtraction can be considered as addition of the negative or simply subtracting element by element Matrix addition is commutative and associative A B B A A B C A B C Scalar Multiplication Multiplying a matrix by a scalar a number means multiplying each element of the matrix by that number 2 A 3 4 1 0 0 7 1 6 5 2 6 4 2 A 6 8 2 0 2 0 10 14 4 12 12 Scalar multiplication of a matrix is associative and distributive k1k2 A k1 k2A 1A A k A B kA kB k1 k2 A k1A k2A Matrix multiplication is only defined if the matrices have compatible size Specifically to multiply AB A must have the same number of columns as B has rows If A is m X n ESE 318 02 Fall 2014 then B must be n X p and the result is m X p This can be extended to multiplying several matrices as follows subscripts denote matrix size Also matrix multiplication is associative and distributive assuming the matrices have compatible sizes Am n Bn o C o p D p q E q r Fm r ABCDE A BCDE AB CDE A B C DE etc A B C AB AC B C A BA CA However matrix multiplication is not commutative in general AB BA The mechanics of matrix multiplication is best seen visually Think of multiplying AB as multiplying rows of A times columns of B 3 A 2 3 C 2 1 7 2 5 6 4 3 1 7 5 1 8 2 B 6 3 5 4 5 3 2



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