Introduction To Matrices MCV 4UI Assignment #1

Transcription

1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be n element or entry of the mtrix DIMENSIONS: The mtrix hs rows nd columns Its dimensions re x The mtrix hs rows nd columns Its dimensions re x The mtrix [ ] hs row nd columns Its dimensions re x The mtrix hs rows nd columns Its dimensions re x

2 In generl, n mxn red m by n) mtrix is mtrix with m rows nd n columns The numbers m nd n re the dimensions of the mtrix mn m m m n n A The element ij in the i th row nd the j th column is clled the i,j) th entry A xn mtrix is lso clled row vector nd mx mtrix is clled column vector EQUAL MATRICES: Two mtrices re sid to be the sme size if they ech hve the sme dimensions Two mtrices re equl if nd only if they hve the sme size nd ll their corresponding elements re equl ZERO MATRIX: A zero mtrix, O, cn be of ny size where ll its entries re zero Exmples: ) O b) O IDENTITY MATRIX: An identity mtrix, I, must be squre mtrix equl rows nd columns) where ll entries re zero except for the digonl which contins ll ones Exmples: ) x I b) x I c) x I MATRIX ADDITION: If A nd B re mtrices of the sme size, their sum A B is mtrix of the sme size, formed by dding corresponding entries in A nd B Exmple: ) ) ) ) Note: The sum does not exist since the two mtrices do not hve the sme size

3 PROPERTIES OF MATRIX ADDITION: If A, B, C nd the zero mtrix, O, re of the sme size then the following properties re vlid A B B A Commuttive Lw) A B C) A B) C Associtive Lw) A O A Ech mtrix, A, hs negtive, -A, such tht A -A) O SCALAR MULTIPLICATION: If A is mtrix nd k is sclr, the product ka is mtrix of the sme size s A in which every entry is multiplied by k Exmple: NOTE: A A A for ny mtrix A PROPERTIES OF SCALAR MULTIPLICATION: If A nd B re of the sme size nd m nd n re rel numbers then the following properties re vlid mna) mn)a ma B) ma mb m n)a ma na A A Associtive Lw) Distributive Lw) Distributive Lw) MATRIX MULTIPLICATION: Not ll mtrix multiplictions re possible In order to multiply mtrices A nd B to get the mtrix AB, the number of columns of A must equl the number of rows of B We cn multiply hs rows nd since the first mtrix hs columns nd the second mtrix Now to multiply mtrices, we multiply ech row of the first mtrix by ech column of the second mtrix ) multiply st row by the st column row, column

4 ) multiply st row by the nd column row, column ) multiply st row by the rd column row, column ) multiply st row by the th column row, column So fter multiplying the first row of the first mtrix by ll the columns of the second mtrix we get: Now we continue by multiplying the second row of the first mtrix by ll the columns of the second ) ) multiply nd row by the st column row, column ) ) multiply nd row by the nd column row, column

5 nd so on multiply nd row by the rd column row, column multiply nd row by the th column row, column In order to multiply mtrix A p r nd mtrix B s t, the number of columns of A hs to equl the rows of B ie r s If r s then the resulting mtrix is AB with dimensions p t PROPERTIES OF MATRIX MULTIPLICATION: If A, B nd C re mtrices of sizes for which the following opertions cn be performed, nd if I is the identity mtrix of the pproprite size, then the following properties re vlid ABC) AB)C AI A IA AB C) AB AC B C)A BA CA Associtive Lw) Identity) Distributive Lw) Distributive Lw) THE TRANSPOSE OF A MATRIX t A ) The trnspose of mtrix is found by interchnging the rows nd columns of mtrix Exmple: If A then t A

6 Introduction To Mtrices EXERCISES Assignment # Instructions: Answer the following questions on seprte sheet of pper Use the following mtrices to nswer the questions below: [ ] A B C D E F G Stte the dimension of ech mtrix bove For mtrix C, stte the element c Stte nother nme for mtrix A Stte nother nme for mtrix D Wht is the nme given to mtrices F nd G? Determine the following if possible If it is not possible explin why ) AB b) CD c) DA d) B-E e) B C f) BF g) EFB h) CG i) GB E j) F k) G l) GC m) E n) G E t o) GB C t