. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry 8. Inverse of x mtrix 9. Applitions: systems Crmer s Rule Prolem Solving Assignment nd nswer key 0. Prtie Exm nd Answer Key
Definition: A mtrix is retngulr rrngement of elements (entries) presented etween rkets [ ] or doule lines Exmples: [ ] g i
Bsi Conepts: Nming of mtrix: pitl letters re used to denote mtrix Mtrix Dimension: the numer of rows (horizontl) y the olumns (vertil) Given: 0 We n refer to this mtrix s: A x whih trnsltes s mtrix A hving dimensions y Note: dimension is lwys given s susript nd the vrile x reds y not times.
Types of mtries: ) row mtrix - ontins only one row [ ] ) olumn mtrix - ontins only one olumn ) zero mtrix - ll entries re zero 0 0 0 0 d) trnspose mtrix - interhnging rows nd olumns A A T
Addition nd Sutrtion of Mtries Just s the sum or differene of two rel numers is unique rel numer, the sum or differene of two mtries is unique mtrix. 0 8 ( ( )) ( 0) ( 8) ( ( )) ( ( )) (( ) ) A B C x x x Only mtries of the sme dimensions n e dded or sutrted. If dimensions re different the ddition or sutrtion is undefined.
Mtrix ddition is oth ommuttive nd ssoitive: Given: 8 A, B C, Using the given informtion prove tht the given sttement is true. A) Commuttive A B B A or B C C B B) Assoitive A ( B C ) (A B) C
O mxn The sum of the zero mtrix nd ny other mtrix is A mxn, the zero mtrix is the identity element for ddition in the set of ny mtrix hving dimensions m x n A mxn Exmple: 0 0 0 0 A mxn The dditive inverse (or negtive) of the mtrix is the mtrix - A mxn whose entries re the negtive of the orresponding entries in A. if A then A,, euse 0 0 0 0
As with rel numer, the onept of sutrtion of mtries n e redefined to s follows: the ddition of n dditive inverse A B A ( B ) mxn mxn mxn mxn 0
Slr Multiplition When we work with mtries we refer to ny rel numer s slr. The produt of slr s nd mtrix A mxn is the mtrix s A mxn eh of whose entries re s times the orresponding entry in A. 0 0 0 0 Exmple: 0 The produt of slr nd mtrix n e shown to e oth ommuttive nd ssoitive. This proof is left up to you.
Assignment: Given: A B [ ] C D E,,,, 9. Determine the dimensions of mtries A, B, nd C. Wht is the zero mtrix for mtrix B?. Wht is the trnspose of mtrix B? of mtrix E?. Wht is the dditive inverse for mtrix A? for mtrix D?. Determine the resultnt mtrix sed on the following opertions: ) x A or A ) - x C or -C ) (A D) E d) A -E e) D -(A E)
. Solve for the vrile mtrix: ) d 8 ) x y u v
Answer Key. A x, B x, C x....) B [ 0 0 0] B T, E T A D, 9 8.) ) d) e).) ) 8 0 0 0 0 8 0 9
Mtrix Multiplition [ ] A x y z B Given: nd with the numers x, y, nd z representing the numer of ssettes, CD s nd DVD s sold eh week t lol retil outlet while the,, nd represent the prie of eh item during regulr sles week nd,, nd represent the prie of eh item during speil promotions week. Prolem: wht were the gross reeipts for the first weeks sles nd wht were they during the speil promotion. Note: Gross mount on one item numer of items x ost per item Gross reeipts totl of gross mounts for the vrious items Reeipts for st week
Gross reeipts for the first week ost of ssettes is x, of CD s y nd of DVD s z or x y z Gross reeipts for the speil promotion ost of ssettes is x, of CD s y nd of DVD s z or x y z The proess of dding the produts otined y multiplying the elements of row in one mtrix y the orresponding elements of olumn in nother mtrix defines the proess of mtrix multiplition
It is importnt to note tht for mtrix multiplition to our: ) the numer of elements in row in the first mtrix must mth the numer of elements in olumn in the seond mtrix ) the resulting nswer will lwys hve dimensions defined y the numer of rows in the first mtrix nd the numer of olumns in the seond mtrix Represent the dimensions of resulting mtrix A x * B x d C x d If equl, mtrix multiplition n our
Generl Notes: Mtrix multiplition differs from tht of rel numers in tht it is not, in generl, ommuttive. 0 nd Therefore, when it is neessry to find produt we must py strit ttention to the order in whih it is written. AB mens leftmultiplition of B y A, nd BA mens right-multiplition of B y A. if A nd B AB nd BA, 8 8
If we hve squre mtrix whose min digonl (from upper left to lower right) onsists of entries of I nd ll other entries re 0, we refer to this s n identity mtrix nd is leled I. The following re exmples of identity mtries for x nd x mtries. 0 0 0 0 nd 0 0 0 0 Whether we use left or right multiplition, the identity mtrix times given mtrix results in the given mtrix. 0 0 0 0
Given: 0 0 0 0 0 0 illustrtes tht even though the produt of two mtries equls zero mtrix does not imply tht the first mtrix must equl zero or tht the seond mtrix must equl zero. If we were to ompre this with the rel numer system nd were given xy0, then either x 0 or y 0 Two vlid lws re left up to you to prove: Use the given mtries nd prove the following: ) The ssoitive lw (AB)C A(BC) ) The distriutive lws: AB AC A(B C) nd BACA (B C)A A B C,, 8
Assignment:... [ ] [ ] - 8.... 9-0 - - 9
Answers:..... [ ] 9 8 8 [ ] - 9. 0 0 8. 9 8 0
The Determinnt Funtion δ The ssoition of rel numer with squre mtries of ny dimension (order) is lled the determinnt of the mtrix. Determinnts n e distinguished from mtrix euse they re lwys enlosed within single set of vertil lines. The entries re lled elements of the determinnt nd the numer of entries in ny row or olumn is lled the order of the determinnt. The vlue of the rel numer defined y the rry is lulted y following definite rules referred to s expnding the determinnt.
To lulte the vlue of the determinnt A) For x determinnt: Given: A d Gret, formul tht mkes it esy Then: δ A d d Exmple #: B δ 8 B ( )( 8) ()( ) Exmple #: ( )( ) ( )()
Assignment: Clulte the vlue of eh x determinnt.. 8.. 8 8
Answer Key:.. -. 8. -8
B. For x determinnt Proedure #- digonl method gol to disply sum of produts tht inludes ll possile rrngements of the susripts of the letters, nd. Given: Copy the first two olumns in order to the right of the rd olumn Multiply eh entry in first row y other two entries in the digonl moving from left to right (these produts re the first three terms of determinnt)
Multiply eh element in the row, strting from the right, y the other entries on the digonl going right to left. The negtives of these produts re the lst three terms of the determinnt ( ) ) ( δ Exmple: A A δ ( ) ( ) ( ) 9 8-0 ) 0 ( 8 8 0 ()()() )()() ( )()() ( )()() ( ()()() )()() ( A δ
Assignment: Clulte the vlue of eh determinnt using the digonl method.... 8
Answer Key:. -00. 0..
Proedure # - Expnsion By Minors: A minor of n element in determinnt is the determinnt resulting from the deletion of the row nd olumn ontining the element. Given: The minor of in is The minor of in is
Steps:. Multiply eh element in the hosen row or olumn y its minor.. Determine whether eh element is to e ssigned or -. If the sum of the numer of the row nd the numer of the olumn ontining the element is odd ssign - nd if even ssign. Add the resulting produts Exmple: ( ) ( ) ( )( ) ( 0 ( )) () ( ( 8)) ( ) (9) ()
Assignment: Determine the vlue of eh determinnt using expnsion y minors. A. Aout indited row or olumn.. B. Aout ny row or olumn.. olumn ; 8 row ; 8 9 0
Answer Key:.-00. -8. -.-89
Proedure #: Using properties of determinnts Property #: You n rete the negtive of n originl determinnt y interhnging ny two rows or olumns. - - nd 9-9 Property #: The determinnt hs vlue of 0 if two rows or olumns re identil 8 0 8 Property#: The determinnt hs vlue of 0 if one row or olumn hs 0 for every element 0 0 0 0
Property #: We n rete determinnt equl to the originl if we interhnge ll the rows nd olumns in order. 0 nd 0 Property #: We n rete determinnt k times the originl if we multiply the elements of one row or olumn y rel numer k. 8 nd 8 8 Multiply st olumn y
And now for the most importnt property Property #: If eh element in ny row or olumn is multiplied y rel numer k nd the resulting produts re dded to the orresponding elements of nother row or olumn, the resulting determinnt equls the originl. 8 Multiply olumn y nd dd to (8) 8 8 olumn () 0 ( ) 9
Why is this so importnt?? This llows the mnipultion of the elements of the determinnt with the gol of mximizing the numer of elements equl to zero whih in turn llows expnsion y minors to fous on only one minor nd x determinnt. Exmple #: ( ) () () 8 Multiply nd row y nd dd to st row 0 0 8 Multiply nd row y nd dd to rd row ( ) () () 0 9 Expnd y minors (8() (9)) 8 ( ) 9 0 8
Exmple #: x- () () ( ) () () () () () () x- x- 0 0 0 0 (( 0) ( )( ))
Assignment Clulte the vlue of the determinnt using the properties of determinnts nd reting zeros in rows nd olumns.. 8 8 9 8 8
Answer Key:.-. 8. -9. 0
Summry out determinnts:. The formul d- works with n y x determinnt. The digonl method only works with x determinnts. Expnsion y minors will work with ny squre mtrix with dimensions three or greter.. The retion of zeros (row nd olumn opertions) provides the esiest pproh to determine the determinnt nd minimize the numer of lultion errors.
The Inverse of x Mtrix Rememer: results in the identity mtrix. When the produt of two rel numers is the multiplitive identity I, the two numers re lled multiplitive inverses. Similrly, ny two mtries A nd B suh tht AB BA I, re lled inverses. To identity the inverse of mtrix A, it is ustomry to sustitute A - for B. 0 0 8 8 8 8 How do we lulte the inverse??
Sure hope we hve formul!! Let u v A A AA I d nd x y suh tht u u u d x x dx v v y v 0 y dy 0 0 0 For the ove sttement to e true: u u x v dx 0 v y dy 0
d 0 If, you n solve these pirs of equtions (sorry, this gret exerise is left up to you) for u nd x, nd for v nd y, respetively, nd find: u x d d d,, v y d d Sine eh denomintor is δ A, we n write A - s: A d δ A In Summry: To get A - ) interhnge the nd d ) hnge the signs on nd ) multiply the resulting mtrix y the determinnt of A.
Exmples: ) Determine the inverse mtrix for the given mtrix: ) Solve for mtrix A ) ()( ()() ; A A 0 0 0 9 0 9 0 0 0 A A A
Assignment: A. Determine the inverse of the following mtries:.. 8 9 8 B. Solve for the mtrix A... 8 A 9 A
Answer Key: 8 A... 0 0 9 0 8 0 B... 9 9 0
Applitions of Mtries: ) Solving Systems of Equtions: y x y x y x Reson - mtrix multiplition If we re given: it is possile to write n equivlent mtrix eqution: nd y x y x y x Whih n e trnslted into: oeffiient mtrix X vrile mtrix onstnt mtrix
To solve: y x Rememer the use of the multiplitive inverse A y x y x δ A ) (, ) ( δ δ A y x ) ( ) ( δ δ Solution Set:
Exmple # Find the solution set for the given system: 9 0 9 9 y x y x y x y x y x Mtrix eqution Multiplitive inverse Solution Set {(/9, -0/9)}
Exmple # Find the solution set for the following prolem. Use mtries. The mesure of one of two omplementry ngles is degrees less thn three times tht of the other. How lrge re the ngles? Let x mesure of lrger ngle in degrees y mesure of smller ngle in degrees Open sentenes: x y 90 x y - To use mtries it is required tht:. Both vriles must e on the left side of the equl sign nd the onstnt on the right side.tht the order of vriles e mintined in oth questions (use order sed on position in lphet)
90 90 y x y x y x Mtrix eqution Multiplitive Inverse Solution Set:{(, )}
B. Crmer s Rule Wht is this? A mtrix proedure tht n e used to solve systems of equtions - espeilly equtions tht hve three or more vriles. To understnd the rule let us first exmine the proedure s it would pply to system of equtions of two vriles y x y x nd Let x x xd D Coeffiient mtrix Property # y x y x xd Apply Property #
If D 0, x D D D x In summry:. Crete oeffiient mtrix D. Crete D x mtrix y repling the x oeffiients in the oeffiient mtrix with the onstnt vlues. The rtio D x /D defines the vlue for the vrile x.. Similrly, rete mtrix D Y y repling the y oeffiients in the oeffiient mtrix with the onstnt vlues. The rtio D y /D defines the vlue for the vrile y.. The replement of oeffiients with onstnts nd the retion of the rtio is repeted equl to the numer of different vriles
Exmple # Determine the solution set for the following system of equtions using Crmer s Rule x - y x y D D D x y ()() ()() ()() ( )() ( )() ()() x D D Dy, y D x Solution Set: {(/,-/)}
Exmple # Determine the solution set for the following system: (Rememer to lulte the vlue of the vrious determinnts you n use the digonl method or row or olumn opertions whih result in zeros) x - y z - -x y - z x y - z D 0 D x 8 Continues on next slide
0 z y D D 0 0, 0, 0 8 D D D D D D z y x Solution Set: {(9/0, /0, -/)}
C. Prolem Solving We wish to lulte the mximum height roket will reh, when will it reh this height, nd when it will hit the ground? A trking sttion ws le to give the ltitude of the roket t vrious horizontl distnes from the lunh site. The olleted dt ws s follows: ltitude of 8 kilometers t distne of kms, n ltitude of kms t 0 kms, nd n ltitude of 0 kms t kms. We will ssume tht the ojet in free flight will hve pth tht pproximtes prol defined y eqution f ( x) x x To use the eqution we will hve to determine the vlues of, nd y sustituting the oordintes of the positions we know we will rete three equtions.
f () f( 0) f () () ( 0) () () () 8 ( 0) or 0 or or 8 00 0 0 To determine the vlues of, nd it is neessry to rete the D, D, D nd D determinnts D 00 D 00 0 8 0, D, D 8 0 0 00 0 8 0 D D D D D D 0.08 8 0
The eqution of the pth of the ojet is: f ( x) 0.08x 8x A) to determine the horizontl distne t whih the roket rehes the mximum height x 8 ( 0.08) 0 B) to lulte the mximum height F (0 ) 0.08 (0 ) 8(0 ) mx height is 00 kilometers mximum height t 0 kms form lunh 00 00 00 C) when the roket hits the ground (the ltitude of the roket will e 0) ± 8± 8 ( 0.08)(0) x ( 0.08) 8± 8 0. 0 nd 00 0 lunh position 00 point of impt
Assignment:. Solve the following system of equtions using the ide of multiplitive inverses. ) x - y ) 9x - y -x y -8 x y -. Solve the following systems of equtions using Crmer s Rule ) x -y - ) -x y z x - y x y - x - x y - z. A roket is trked t three distint points in flight. The position is reorded s pirs of the form (distne from lunh pd, ltitude). Find mximum height nd distne to impt from lunh site. (, 8), (, ), nd (, )
0. ) {(, )}. ) {(, )}, ) {( )} 9 9,, ) {( )}. Mximum height 0 distne to impt 0
Prtie exm: Given : ) A B C ) A B ) C T d) A*B C e) B - f) A..Solve: ) ) x y 9 -x y -, 8, C B A d δ
) x y z - -x y z x y z -..Multiply the following: ) ) 8 0
.Determine the vlue of the determinnt using the indited method: ) digonl ) expnsion y minors ) properties of determinnts 8 8 8 8
Answers:. ) 0 ) ) d) 0 8 e) f) - 8 9 0. A) ) {(, )} ) {(,, )}. ) 8 ) 9 8. ) ) - ) -8