Matrices and vectors

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1 Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra However, for those whose kowledge may be rusty from log disuse or raw with recet acquisitio, sectios o the ecessary material from these subjects have bee icluded where appropriate Although these revisio sectios (marked with the symbol are as self-cotaied as possible, they are ot suitable for those who have o acquaitace with the topics covered The material i the revisio sectios is surveyed rather tha explaied It is suggested that readers who feel fairly cofidet of their mastery of this surveyed material sca through the revisio sectios quickly to check that the otatio ad techiques are all familiar before goig o Probably, however, there will be few readers who do ot fid somethig here ad there i the revisio sectios which merits their close attetio The curret chapter is cocered with the fudametal techiques from liear algebra which we shall be usig This will be particularly useful for those who may be studyig liear algebra cocurretly with the preset text Algebraists are sometimes eglectful of the geometric implicatios of their results Sice we shall be makig much use of geometrical argumets, particular attetio should therefore be paid to 13 owards, i which the geometric relevace of various vector otios is explaied This material will be required i Chapter 3 Those who are ot very cofidet of their liear algebra may prefer leavig 110 util Chapter 5 11 Matrices A matrix is a rectagular array of umbers a otatio which eables calculatios to be carried out i a systematic maer We eclose the array i brackets as i the examples below: B ( 1 2 A matrix with m rows ad colums is called a m matrix Thus A isa3 2 matrix ad B isa2 3 matrix 1 wwwcambridgeorg

2 Chapter 1 Matrices ad vectors See Chapter 13 A geeral m matrix may be expressed as c 11 c 12 c 1 c 21 c 22 c 2 C c m1 c m2 c m where the first umber i each subscript is the row ad the secod umber i the subscript is the colum For example, c 21 is the etry i the secod row ad the first colum of the matrix C Similarly, the etry i the third row ad the first colum of the precedig matrix A ca be deoted by a 31 : colum a 31 3 row We call the etries of a matrix scalars Sometimes it is useful to allow the scalars to be complex umbers but our scalars will always be real umbers We deote the set of real umbers by R Scalar multiplicatio Oe ca do a certai amout of algebra with matrices ad uder this ad the ext few headigs we shall describe the mechaics of some of the operatios which are possible The first operatio we shall cosider is called scalar multiplicatio IfA is a m matrix ad c is a scalar, the ca is the m matrix obtaied by multiplyig each etry of A by c For example, Similarly, ( 1 5B 5 2 Matrix additio ad subtractio ( If C ad D are two m matrices, the C + D is the m matrix obtaied by addig correspodig etries of C ad D Similarly, C D is the m matrix obtaied by subtractig correspodig etries For example, if the C C + D ad D wwwcambridgeorg

3 11 Matrices ad C D Note that C + C C ad that C C The fial matrix is called the 3 3 zero matrix We usually deote ay zero matrix by 0 This is a little aughty because of the possibility of cofusio with other zero matrices or with the scalar 0 However, it has the advatage that we ca the write C C 0 for ay matrix C Note that it makes o sese to try to add or subtract two matrices which are ot of the same shape Thus, for example, A + B is a etirely meaigless expressio + ( 1 2 Matrix multiplicatio If A is a m matrix ad B is a p matrix, m a 11 a 12 a 1 a 21 a 22 a 2 B p b 11 b 12 b 1p b 21 b 22 b 2p a m1 a m2 a m b 1 b 2 b p the A ad B ca be multiplied to give a m p matrix AB: A B AB To work out the etry c jk of AB which appears i its jth row ad kth colum, we require the jth row of A ad the kth colum of B as illustrated below 3 wwwcambridgeorg

4 Chapter 1 Matrices ad vectors p b 1k b 2k p m a j1 a j2 a j m c jk b k The etry c jk is the give by c jk a j1 b 1k + a j2 b 2k + a j3 b 3k + +a j b k Example 1 We calculate the product AB of the matrices ( B Sice A isa2 3 matrix ad B isa3 2 matrix, their product AB isa2 2 matrix: ( ( AB a b c d To calculate c, we require the secod row of A ad the first colum of B These are idicated i the matrices below: a b c d We obtai Similarly, c Thus a b d AB ( ( Note agai that it makes o sese to try to calculate AB uless the umber of colums i A is the same as the umber of rows i B Thus, for example, it makes o sese to write wwwcambridgeorg

5 11 Matrices Idetity matrices A matrix is called a square matrix for obvious reasos Thus, for example, is a square matrix The mai diagoal of a square matrix is idicated by the shaded etries i the matrix below: a 11 a 12 a 1 a 21 a 22 a 2 a 1 a 2 a Note that a idetity matrix must be square Just as a zero matrix behaves like the umber 0, so a idetity matrix behaves like the umber 1 The idetity matrix is the matrix whose mai diagoal etries are all 1 ad whose other etries are all 0 For coveiece, we usually deote a idetity matrix of ay order by I The 3 3 idetity matrix is I Geerally, if I is the idetity matrix ad A is a m matrix, B is a p matrix, the AI A ad IB B Example 2 (i (ii ( ( There is some risk of cofusig this otatio with Determiats the modulus or absolute value of a real umber I fact, the With each square matrix there is associated a scalar called the determiat of the determiat of a square matrix may be positive or matrix We shall deote the determiat of the square matrix A by det(aorby A egative A1 1 matrix (a is just a scalar, ad det( a 5 wwwcambridgeorg

6 Chapter 1 Matrices ad vectors The determiat of the 2 2 matrix ( a b c d is give by det( a c b d ad bc To calculate the determiat for larger matrices we eed the cocepts of a mior ad a cofactor The mior M correspodig to a etry a i a square matrix A is the determiat of the matrix obtaied from A by deletig the row ad the colum cotaiig a I the case of a 3 3 matrix a 11 a 12 a 13 a 21 a 22 a 23 a 31 c 32 a 33 we calculate the mior M 23 correspodig to the etry a 23 by deletig the secod row ad the third colum of A as below: a 11 a 12 a 13 a 21 a 22 a 23 a 31 c 32 a 33 The mior M 23 is the the determiat of what remais ie M 23 a 11 a 12 a 31 a 32 a 11a 32 a 12 a 31 If we alter the sig of the mior M of a geeral matrix accordig to its associated positio i the checkerboard patter illustrated below, the result is called the cofactor correspodig to the etry a I the case of a 3 3 matrix A, the cofactor correspodig to the etry a 23 is equal to M 23, sice there is a mius sig,, i the secod row ad the third colum of the checkerboard patter: colum row wwwcambridgeorg

7 11 Matrices The determiat of a matrix A is calculated by multiplyig each etry of oe row (or oe colum by its correspodig cofactor ad addig the results The value of det(a is the same whichever row or colum is used Example 3 The determiat of the 1 1 matrix (3 is simply det( 3 The determiat of the 2 2 matrix ( is det( We fid the determiat of the 3 3 matrix i two ways: first usig row 1 ad the usig colum 2 The etries of row 1 are a 11 1 a 12 2 a 13 3 We first fid their correspodig miors, M M M ad the alter the sigs accordig to the checkerboard patter to obtai the correspodig cofactors, +M 11 5 M 12 ( 1 1 +M 13 7 The det( a 11 (M 11 + a 12 ( M 12 + a 13 (M 13 1( 5 + 2(1 + 3( We calculate the determiat agai, this time usig colum 2: a 12 2 a 22 1 a wwwcambridgeorg

8 Chapter 1 Matrices ad vectors The correspodig cofactors are M M M (2 9 7 Thus 5 det( a 12 ( M 12 + a 22 (M 22 + a 32 ( M 32 2(1 + 1( 5 + 3( A ivertible matrix is also called osigular If A is a matrix, the A 1 is a matrix as well (otherwise the equatio would make o sese Iverse matrices We have dealt with matrix additio, subtractio ad multiplicatio ad foud that these operatios oly make sese i certai restricted circumstaces The circumstaces uder which it is possible to divide by a matrix are eve more restricted We say that a square matrix A is ivertible if there is aother matrix B such that AB B I I fact, if A is ivertible there is precisely oe such matrix B which we call the iverse matrix to A ad write B A 1 Thus a ivertible matrix A has a iverse matrix A 1 which satisfies It ca be show that AA 1 A 1 I A is ivertible if ad oly if det(a 0 If A is ot square or if A is square but its determiat is zero (ie A is ot ivertible, the A does ot have a iverse i the above sese Alterative otatios for the traspose are A ad A t Traspose matrices I describig how to calculate the iverse of a ivertible matrix, we shall eed the idea of a traspose matrix This is also useful i other coectios If A is a m matrix, the its traspose A T is the m matrix whose first row is the first colum of A, whose secod row is the secod colum of A, whose third row is the third colum of A ad so o m a 11 a 12 a 1 a 21 a 22 a 2 a m1 a m2 a m A T m a 11 a 21 a m1 a 12 a 22 a m2 a 1 a 2 a m 8 wwwcambridgeorg

9 11 Matrices A importat special case occurs whe A is a square matrix for which A T Such a matrix is called symmetric (about the mai diagoal Example 4 (i If , the A T ( Note that (ii If (A T T ( , the A T Thus A T ad so A is symmetric T A The cofactor method for A 1 The iverse of a ivertible matrix A ca be calculated from its determiat ad its cofactors I the case of 1 1 ad 2 2 matrices, oe might as well lear the resultig iverse matrix by heart A1 1 ivertible matrix (a is just a ozero scalar ad ( 1 A 1 provided a 0 a A2 2 ivertible matrix ( a b c d is oe for which det( ad bc 0 Its iverse is give by ( A 1 1 d b ad bc c a The formulas ca easily be cofirmed by checkig that AA 1 ad A 1 A actually are equal to I For example i the 2 2 case ( ( A 1 1 d b a b ad bc c a c d ( ( 1 da bc 0 I ad bc 0 bc + ad 0 1 The cofactor method gives a expressio for the iverse of a ivertible matrix A i terms of its cofactors 9 wwwcambridgeorg

10 Chapter 1 Matrices ad vectors I the case of a ivertible 3 3 matrix for which det(a 0 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 the iverse is give by A 1 1 det( A +M 11 M 12 +M 13 M 21 +M 22 M 23 +M 31 M 32 +M 33 For a geeral ivertible matrix a 11 a 12 a 1 a 21 a 22 a 2 a 1 a 2 a we have +M 11 M 12 ( 1 1 M 1 A 1 1 M 21 +M 22 ( 1 M 2 det( A ( 1 1 M 1 ( 1 M 2 +M Thus, each etry of A is replaced by the correspodig mior ad the sig is the altered accordig to the checkerboard patter to obtai the correspodig cofactor The iverse matrix A 1 is the obtaied by multiplyig the traspose of the result by the scalar (det(a 1 T T Example 5 We calculate the iverses of the ivertible matrices of Example 3 (i (3, det( 3 0 A 1 ( 1 3 ( 1 2 (ii, det( A 1 1 ( ( (iii , det( 18 0 We begi by replacig each etry of A by the correspodig mior ad obtai wwwcambridgeorg

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:

(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row: Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig