Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED MATIX.. Septemer 6
. DEFINITION OF A MATIX MATIX OPEATIONS A mtrix is simply retngulr rry of numers. Some typil exmples: 6. 8 /. A mtrix is usully denoted y pitl letter, sy 6. A= B= C= 8 /. The first mtrix ove hs rows nd olumns. We sy tht A is x ( two y three ) mtrix, or otherwise tht the order of A is x. Likewise, B is x mtrix while C is x mtrix. The generl form of x mtrix is A= (notie tht for exmple is the element in row nd olumn ) SQUAE MATICES No of rows = No of olumns The order of squre mtrix is nxn, eg. x, x, x et. (in this se we my lso sy: squre mtrix of order n. For exmple,
6 re squre mtries of order nd respetively. We lso sy tht the elements,, (indited ove) form the min digonl of the squre mtrix. OW MATICES Mtries of order xn For exmple A=( ) is x mtrix B=( 6-6) is x6 mtrix COLUMN MATICES For exmple Mtries of order mx A= - is x mtrix B= is x mtrix Notie: Mtries of order x re lso defined, for exmple C=(). THE ZEO MATIX All elements re The x zero mtrix is = The x zero mtrix is O = nd so on!
EQUAL MATICES: A=B A=B if A nd B hve the sme order the orresponding elements re equl EXAMPLE x A= B= y C= s t u D= v w It nnot e A=B sine A is x while B is x A=C implies y=, x=, = (= holds nywy!) B D sine (lthough oth re x) Let A nd B hve the sme order. We define some new mtries: THE SUM A+B we simply dd the orresponding elements THE DIFFEENCE A-B we simply sutrt the orresponding elements THE SCALA PODUCT na (n is slr, i.e. numer) we simply multiply eh element of A y n
EXAMPLE Then A= B= A+B= 6 - A-B= - - - B-A= - - 6 9 A= - -6 -A= - - -9 - - -A=-A= - - - - - / / A= / /.. or.a=.. Finlly, A+B = + 6 9 = + 8 6 9 9 6 = EXAMPLE Let A= nd B=. Find the mtrix X if - A+X=B Method : X must e of size x. Suppose tht X= d Then A+X=B + = d -
+ 6 8 6 = d - + + 6 = 6+ 8+ d - Hene, +=6 = 6+= =9 += =- 8+d=- d=- Therefore, - X= 9 - Method : We solve for X (s usul eqution) A+X=B X=B-A X= - - 6 X= - - 6 8 - X= 9 - The following opertion is the most importnt one for mtries THE PODUCT OF TWO MATICES: AB =? 6 It is NOT s someone would expet!! Here, we do not multiply the orresponding elements.
Let us strt y SPECIAL CASE. We multiply row-mtrix y olumn-mtrix: ( ) x y z = (x+y+z) Notie tht we multiply the orresponding elements nd dd the results. For exmple, ( ) = (. +. +. )=() Now we re redy to multiply two mtries A nd B in generl. First of ll, the orders of A nd B must e s follows: A B m x k k x n Tht is, numer of olumns of A = numer of rows of B. The order of the new mtrix AB will e m x n For exmple, order of A order of B order of AB x x8 x8 x x x x x x x x not defined The multiplition tkes ple s follows: we multiply rows of A y olumns of B (row i) x (olumn j) will give the element ij Ok, it seems omplited!!! elx, tke it esy!!!! Look t the following desription: 6
Let x A= B= y 6 z Notie tht the order of AB is expeted to e x For the first row of AB: multiply row of A y eh olumn of B seprtely: AB = 6 x y z + + = + + x+ y+ z = For the seond row of AB: multiply row of A y eh olumn of B seprtely: AB = 6 x y z + + x+ y+ z = + + = + + + + x+ y+ z x+ y+ 6z EXAMPLE A= 6 B= Then AB= 6 = sine. +. +. =. +. +. = = 9 sine. +. +6. =9. +. +6. =
EXAMPLE Consider gin the mtries A nd B ove. Let us find the produt BA. Notie tht the order is expeted to e x. We otin BA= 6 + = + + 8 + 6+ + 6+ 9+ 8 + = 9 9 6 NOTICE: In generl AB BA They my e of different order, or perhps only one the produts ould e defined (eg. if A is x nd B is x) Even if oth mtries A nd B re squre mtries, sy x, the resulting x mtries AB nd BA re not equl in generl. EXAMPLE 6 Let A= nd B=. Then AB= = while BA= 6 = 6 8 Sometimes though, it hppens AB=BA. Then we sy tht mtries A nd B ommute. 8
9 EXAMPLE Let A= nd B= 6. Then AB= 6 = 6 9 6 nd BA= 6 = 6 9 6 The following squre mtrix plys key role in our theory. THE IDENTITY MATIX I Nmely, I= or I= or I= et EXAMPLE 8 Let A= d nd I=. Then AI= d = d =A nd IA= d = d =A In generl, for ny mtrix A (provided tht the orders of A nd I re pproprite!) In other words, the identity mtrix I plys the role of (unit) when we multiply mtries! in the min digonl elsewhere AI=A nd IA=A
. THE DETEMINANT deta THE INVESE A - x DETEMINANT Let A=. The determinnt of A is the numer d deta= d- It is lso denoted y det =d- d or =d- d EXAMPLE Let A= Let B=. Then deta= -. Then detb= -6 = = - -6 = + = EXAMPLE Solve the eqution x - = x- It is x(x-)+= x -x+= x= or x= x DETEMINANT Let A=. The determinnt of A is the numer deta = + + Ok, I know, it looks horrile!!!!
A more elegnt wy to estimte deta is - + NOTICE: We multiply the elements of row,,, y three little determinnts respetively. For, the orresponding x determinnt n e otined if we eliminte the row nd the olumn of Similrly for nd. Notie lso tht the signs lternte. EXAMPLE Let A= 6. Then 8 6 deta = - + 8 8 6 =. -. +. =- The following terminology is lso used If deta= we sy tht the mtrix A is singulr. If deta we sy tht the mtrix A is non-singulr.
EXAMPLE Let A= x y z B= x y z C= Mtries like A re known us upper-tringulr Mtries like B re known us lower-tringulr Mtries like C re known us digonl We n esily verify tht deta=.. detb=.. detc=.. For exmple, = = -8 = = THE INVESE A - OF A X MATIX Let A=. The inverse of A is new mtrix given y d A - = deta d - - It is defined only if deta. EXAMPLE Let A= Let B= 6. Then deta= nd the inverse mtrix is A - -6 - = = - -/ 8. Then detb= nd hene, B - is not defined.
Let us multiply the mtries A nd A - of the exmple ove. AA - = A - A= -/ 6 -/ - - = =I 6 = =I This is not identl! In generl AA - =I nd A - A=I NOTICE: Compre with numers NUMBES The inverse of numer is - - = nd - = Is ny numer invertile? NO. Only if (if =, - is not defined) If is invertile then MATICES The inverse of mtrix A is A - AA - =I nd A - A=I Is ny Mtrix invertile? NO. Only if deta (if deta=, A - is not defined) If A is invertile then - = A - = deta d - - NOTICE If AB=I or BA=I we know tht A is invertile nd B is the inverse of A, tht is A - = B (B is lso invertile nd B - = A).
THE INVESE A - OF A X MATIX The expliit formul for A - is out of our sope! It is enough to know tht A - exists only if deta AA - =I nd A - A=I If AB=I or BA=I then B is the inverse of A A - my e found y lultor (GDC) EXAMPLE 6 Let A=, B= - - - C= ) Find AB nd BA ) Find the inverse of A ) Find the inverse of B Solution ) We n esily see tht AB=I nd BA=I ) Clerly A - =B (the GDC lso gives the sme result) ) Similrly B - =A (the GDC lso gives the sme result) Notie tht we nnot divide mtries, for exmple B/A is not defined. However, we n multiply B y A -, either s BA - or s A - B, ording to the sitution. The following exmple will e hrteristi:
EXAMPLE 6 Let A= 6 nd C= 8. Find B given tht AB=C 8 Unfortuntely, we nnot sy tht B=C/A Method : (nlytil) Let B= 6. Then AB= d + 6 + 6d = d + + d Then AB=C implies Tht is, + 6 + +6= += + 6d 8 = + d 8 +6d=8 +d=8 The first two equtions give =, = The seond two equtions give =, d=. Therefore, B=. A muh more prtil method is the following Method : (solve for B) The mtrix A is invertile with A - = -/ -. Thus, we my multiply oth prts of AB=C y A - on the left AB=C A - AB=A - C IB= A - C B=A - C = -/ - 8 = 8
EQUATIONS WITH MATICES Suppose tht squre mtries A,B,C re known. Find the unknown mtrix X in eh of the following equtions: Mtrix eqution Solution A+X=B A+X=B -A+X=B X=B-A X=B-A X=A+B X=/(A+B) AX=B X=A - B (mind the order!) XA=B X=BA - (mind the order!) AXB=C X=A - CB - AX+B=A AX=A-B X=A - (A-B) X=I-A - B XA-B=XC XA-XC=B X(A-C)=B X=B(A-C) - AX+X=B (A+I)X=B X=(A+I) - B Let us solve in detil some of the ove equtions: EXAMPLE Let A= nd B=. Solve the equtions ) -A+X=B ) AX =B ) XA=B d) AX+B=A e) AX+X=B 6
Solution ) -A+X=B X = /(A+B) X = / 8 9 / / = 9/ ) AX =B X = A - B X = - X = - - ) XA =B X = BA - X = X = - - - - d) AX+B=A X = I-A - B X = X = - - - - e) AX+X=B X = (A+I) - B X = X = X = - - 6 6/ = /
. SYSTEMS OF LINEA EQUATIONS THE FOM AX=B Consider the system x+y=9 x+y=9 If we solve it in the trditionl wy (or y GDC) we will relize tht x= nd y= (hek!) Let A= x X= y 9 B= 9 the mtrix of oeffiients the mtrix of unknowns the mtrix of onstnts The eqution of mtries AX=B is equivlent to x 9 x+ y =, tht is y 9 9 = x+ y 9 whih gives in ft the system of liner equtions ove! Consider now the system x+y-z = - x +y -z = - x -y +z = Agin, if A= - x X= y z - B= - the system n e written in the form AX=B 8
In generl, ny system of n liner equtions nd n unknowns n e expressed in the form AX=B Hene, if A is invertile (tht is if deta ) the solution is given y X=A - B EXAMPLE Consider the x system given ove x+y=9 x+y=9 whih n e written in the form AX=B. Sine deta= =, A - = - - / -/ = - The solution of the system is given y / X = A - B = - -/ 9 = 9 In other words, x= nd y=. EXAMPLE Consider the x system given ove x+y-z = - x +y -z = - x -y +z = whih n e written in the form AX=B. 9
Find the solution of the system, given tht A - = / / 8/ 9/ 8/ / / / / The solution of the system is given y / X = A - B = / 8/ 9/ 8/ / /- / - / = - EXAMPLE Let A=, B= - -, C=, D= - d) Find AB e) Write down the system of three liner equtions whih orresponds to the mtrix eqution BX=C f) Solve the system of liner equtions BX=C g) Solve the mtrix eqution BX=D (this is not system) Solution ) We esily otin AB=I. Hene, A nd B re inverse to eh other. ) y -z = -y +z = x+y-z = ) BX=C X=B - C X=AC X= 8 X= Tht is x=8, y=, z= d) BX=D X=B - D X=AD X= 8 X= 8
Notie though tht if A is not invertile (A - does not exist) the system AX=B nnot e solved in this wy. In generl, for the system AX=B deta (A - exists) deta= (A - doesn t exist) UNIQUE SOLUTION X=A - B NO SOLUTION, or INFINITELY MANY ( ) SOLUTIONS EMAK. Compre with numers. For the eqution x= ( - = exists) Unique solution x=. No solution = ( - doesn t exist) (e.g. x= hs no solution) Infinitely mny solutions (e.g. x=, true for ny xє) Let us see two exmples of systems of liner equtions, where deta= EXAMPLE Consider the systems () x+y= () x+y= x+y= x+y= For oth deta= =, so the systems hve either no solution or n numer of solutions.
We multiply the first solution y nd otin x+y=. Hene the two systems tke the equivlent form () x+y= () x+y= x+y= x+y= System ( ) hs no solution (impossile) System () redues to just one eqution: x+y=. There re infinitely mny solutions: We solve for x: x=-y, y (free vrile). [for severl vlues of y we otin different solutions, for exmple (,), (-,), (-,), (,-), et] For x systems with deta=, only Mth HL needs to go further (next setion!)
. THE AUGMENTED MATIX In this setion we investigte further the system of liner equtions AX=B Let us present first n lterntive wy of getting the unique solution, in se deta USE OF DETEMINANTS Consider the system x+ y= x+ y= Set D=detA= D x = D y = onstnts in the olumn of x onstnts in the olumn of y If D, the unique solution is x= D D x, y= D D y Proof: The system n e written in the form AX=B. Notie tht Thus, the unique solution is A - = D. X= A - B= D = D - + = D D x D = D y D x y /D /D And hene the result!
EXAMPLE Consider gin the x system given in Exmple of the previous setion: x+y=9 x+y=9 We hve D= 9 =, D x = 9 =6 D y = 9 = 9 Therefore, x= D D x = 6 =, y= D D y = = s expeted! Similrly, for x system x+ y+ z=d x+ y+ z=d x+ y+ z=d the determinnts D, D x, D y, D z re defined in n nlogue wy nd if D the unique solution is x= D D x, y= D D y, z= D D z EXAMPLE Consider gin the x system given in Exmple of the previous setion x+y-z = - x +y -z = - x -y +z =
Then D= =- - - D x = - =-, D y = - =, D z = =- nd the unique solution is x= D D x =, y= D D y =-, z= D D z = s expeted! Let us ollet the informtion we hve until now. METHODOLOGY FO X AND X SYSTEMS: AX=B ) We find D=detA ) If D there is UNIQUE SOLUTION, given y X=A - B or in detil y D x= x D y, y=, D D ) If D= we sy tht the system hs either NO SOLUTION NUMBE OF SOLUTIONS For x systems, it is esy to investigte the solution (look t Exmple in previous setion) For x systems, we will use the method of the ugmented mtrix desried elow
Consider the system Here, x + y -z = x +y +z = x +y +z = D=detA= = Hene, there is either NO SOLUTION or AN INFINITE NUMBE OF SOLUTIONS. STEP : Use eqution to eliminte x from equtions nd : x + y - z = y + z = - y +z = - [Equ x Equ] [Equ x Equ] STEP : Use eqution to eliminte x from eqution : x + y - z = y + z = - = - [Equ x Equ] The rd eqution implies tht there is NO SOLUTION. In ft, the unknowns x,y,z do not ply ny role during this proess! We n work only with oeffiients pled in n ugmented mtrix s follows: (row ) (row ) (row ) Then we proeed, step y step, to equivlent ugmented mtries y performing pproprite row opertions. The equivlene etween two mtries is denoted y the symol ~ : 6
~ - - - - ~ - - - The lst row implies tht the system hs NO SOLUTION s it orresponds to the eqution x+y+z=- In generl, the row opertions we my perform in order to otin equivlent mtries re the following METHODOLOGY FO X SYSTEMS WITH deta= ) We onsider the ugmented mtrix of the system d d d ) We trnsform to equivlent mtries of the form y using row s guide nd then y using row s guide Interhnge rows (e.g. ) Multiply row y slr (e.g. ) Add to row the multiple of nother row (e.g. ± )
If the system hs either NO SOLUTION or INFINITELY MANY SOLUTIONS, we expet n equivlent mtrix of the form ) ) if d the system hs NO SOLUTION ) if d=, the system hs INFINITELY MANY SOLUTIONS: We eliminte the lst row of the ugmented mtrix nd trnsform it into the form d (y using row s guide). The mtrix d d implies in ft tht x+ z= d y+ z= d We finlly sy tht the solution is x =d -z y = d -z z (free vrile) EMAK Idelly, in step we ttempt to hve mtries of the form nd Tht is, the leding oeffiient of the guide row should e. EXAMPLE Consider the system x +y +z = x + y -z = x +y +z = Sine deta= we use the ugmented mtrix: 8
9 - ~ ~ - - - - ~ - - - Hene the system hs no solution. EXAMPLE Consider the system x +y +z = x + y -z = x +y +z = Sine deta= we use the ugmented mtrix: - ~ ~ - - - - ~ - - Hene the system hs infinitely mny solutions. We rry on y onsidering - - ~ - -6 Hene, x-6z = y-z = - nd finlly x = + 6z y = - + z z (free vrile)