# UNIT FIVE DETERMINANTS

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1 UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit hs vriety of pplictios. The vlue of the determit of squre mtrix A c e used to determie whether A is ivertile or oivertile. A explicit formul for A exists tht ivolves the determit of A. Some systems of lier equtios hve solutios tht c e expressed i terms of determits.. DEFINITION OF THE DETERMINANT Recll tht i chpter oe the determit of the mtrix A ws defied to e the umer d tht the ottio det A) or A ws used to represet the determit of A. For y give mtrix A [ ij ], the ottio A ij will e used to deote the ) ) sumtrix otied from A y deletig the i th row d the j th colum of A. The determit of y size squre mtrix A [ ij ] is defied recursively s follows. Defiitio of the Determit Let [ ij ] A e mtrix. ) If, tht is A [ ], the we defie det A). ) If >, we defie deta) -) deta ) Exmple If A [], the y prt ) of the defiitio of the determit, det A). If A, the y prts ) d ), det A) ) )det[] ) )det[] ))) ))) If A, the usig prts ) d ), we clculte the det A) s follows.

2 )det ) )det ) )det ) deta) )) ) )) ) )) ). ofctor If A is squre mtrix, the ij th cofctor of A is defied to e ) ij deta ij ). The ottio ij will sometimes e used to deote the ij th cofctor of A. Exmple Let A. The ) det ) ), ) det ) ) d ) det ) ). I the defiitio of the determit, prt ) cosists of multiplyig ech first row etry of A y its cofctor d the summig these products. For this reso it is clled first row cofctor expsio. Exmple Let A. Use first row cofctor expsio to evlute deta). Solutio deta) )det ) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )) )det ) )det ) )det ) )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){)) ) )) ) )) )} )){ } )){ } )){ } )){ }

3 Although the defiitio of the determit uses first row cofctor expsio, the determit of A my e clculted y tig y row or colum) d multiplyig the etries of tht row or colum) y their cofctors d summig the products. This result is give i the ext theorem whose proof is omitted. Theorem Let A e squre mtrix, the i j deta) -) idetai ) ) jdeta j ). i th row cofctor expsio j th colum cofctor expsio Exmple Let A. Evlute det A) y ) secod row cofctor expsio. ) third colum cofctor expsio. Solutio ) det A) ) )det ) )det ) )det )) ) )) ) )) ). ) det A) ) )det ) )det ) )det ))) )) ) )) ). Theorem If A is squre mtrix cotiig row or colum) of zeros, the deta). Proof Use cofctor expsio log the row or colum) of zeros. Theorem If A is mtrix with two ideticl rows or colums), the det A). Proof The theorem is certily true for sice det. If, use cofctor expsio log the row differet from the two ideticl rows. Let this row e the th row. Usig cofctor expsio log this row gives det A) ) )deta ) ) )deta ) ) )deta ). But ech of the sumtrices A, A d A hs two ideticl rows so their determits re, hece det A) for y mtrix. If > proceed s ove writig det A) s sum of products ivolvig sumtrices with two ideticl rows whose determits re.

4 Trigulr d Digol Mtrices A squre mtrix is sid to e upper trigulr mtrix if ll the etries elow the mi digol re zero. A squre mtrix is sid to e lower trigulr mtrix if ll the etries ove the mi digol re zero. A squre mtrix is sid to e digol mtrix if ll etries ot o the mi digol re zero. A digol mtrix is oth upper trigulr d lower trigulr. Exmple is upper trigulr mtrix. is lower trigulr mtrix. is digol mtrix. It is oth upper d lower trigulr. Theorem If A is upper trigulr, lower trigulr or digol, deta). Proof Suppose A is upper trigulr. To evlute deta) use cofctor expsio log the first colum. Sice there is oly oe ozero etry i the first colum the expsio gives deta) -) deta ) deta ). Now A is upper trigulr so proceed s ove to use cofctor expsio log its first colum to get det A ) deta ) where A is A with its first row d first colum deleted. omiig the results gives deta) deta ). otiuig i this fshio, we evetully get det A) s required. If A is lower trigulr or digol, the rgumet is similr. Exmple det ))). det ))) Theorem deti ) for ll. Proof Sice I is digol mtrix, deti ) )) ) Bset-Weve Method The followig method is ltertive wy to evlute the determit of mtrix. This method is oly pplicle to mtrices d is sometimes clled the set-weve method. ostruct rry y writig dow the etries of the mtrix d the repetig the first two colums. lculte the products log the six digol lies show i the digrm. The determit is equl to the sum of products log digols leled, d mius the sum of the products log the digols leled, d.

5 Exmple Use the set-weve method to clculte the determit of A. Solutio det A) ) ) ) ). PROBLEMS. Use the defiitio of the determit to evlute the determit of the give mtrix. ) ) c) d) e) f) g) h). Use the set-weve method to evlute the determits ), ), c) d d).. Evlute the determits of the followig mtrices y ispectio. ) ) c) d). Let. A Fid the followig cofctors of A. ) ) c) d) e) f). Fid ll vlues of for which - det.

6 . ELEMENTARY ROW OPERATIONS ON DETERMINANTS The evlutio of the determit of mtrix usig the defiitio ivolves the summtio of! terms, ech term eig product of fctors. As icreses, this computtio ecomes too cumersome d so other techique hs ee devised to evlute the determit. This techique uses the elemetry row opertios to reduce the mtrix to trigulr form. The effect of ech elemetry row opertio o the vlue of the determit is te ito ccout d the the determit of the trigulr mtrix is evluted y fidig the product of the etries o the mi digol. Theorem If A d B re squre mtrices d B is otied from A y iterchgig two rows or colums) of A, the det B) det A). Proof Let A d B e mtrices. If, the A d B so detb) ) deta). If, the we use cofctor expsio for B log the row tht ws ot iterchged. Let this e row. The detb) ) detb ) ) detb ) ) detb ). Ech sumtrix B j is the sumtrix A j with its rows iterchged so detb j ) deta j ). Hece detb) ) )deta ) ) )deta ) ) )deta ) ) [ ) deta ) ) deta ) ) deta )] ) deta) det A) s required. If > proceed s ove usig cofctor expsios log rows tht were ot iterchged to get the fil result. The proof for iterchged colums is similr. Theorem Let A d B e mtrices with B otied from A y multiplyig ll the etries of some row or colum) of A y sclr. The det B) det A). Proof Suppose B is otied from A y multiplyig the etries of the j th row of A y. Use cofctor expsio for B log its j th row to evlute the determit of B d otig tht these cofctors for B re equl to the correspodig cofctors for A we get det B) ) j j detb j ) ) j j detb j ) ) j j detb j ) ) j j deta j ) ) j j deta j ) ) j j deta j ) [ ) j j deta j ) ) j j deta j ) ) j j deta j )] deta). The proof i the cse where B is otied from A y multiplyig colum of A y is similr. is commo fctor i row of the give mtrix. Exmple det det ) ) orollry det A) det A). Proof Sice ll rows of A re multiplied y the sclr to get A, usig the ove theorem times gives det A) ) ) ) det A) det A).

7 is fctor i row d row Exmple det )) det det ) ) )) Exmple Let A e mtrix d let deta). Fid deta). Solutio deta) deta) )) sice mtrix hs ideticl rows Exmple det det )). Notice tht row of the origil mtrix is times row. This leds to the followig corollry. orollry If A is squre mtrix tht hs row or colum) tht is sclr multiple of other row or colum), the det A). Proof Suppose the j th row of A is times the i th row of A. The det A) det Â ) where Â is the mtrix A with the j th row multiplied y /. But Â hs two ideticl rows, row i row j), so det Â ). Hece det A) det Â ). Theorem Let A e squre mtrix d let B e the mtrix otied from A y ddig multiple of oe row or colum) of A to other row or colum) of A. The det B) det A). Proof Suppose B is otied from A y ddig c times row i to row j. Evlute det B) usig cofctor expsio log row j. The detb) j j ) jdetb j ) ) c i j )detb j ) c i j )deta j ) sice B j A j for ll. j j c ) i deta j ) ) j deta c det Â ) det A) where Â is otied from A y replcig the j th row of A y its i th row. c det A) det Â ) ecuse Â hs two ideticl rows det A) j ) j )

8 Exmple Evlute det Solutio det ))det ))det )det ))))). The previous exmple outlies efficiet techique usig elemetry row trsformtios to evlute the determit of squre mtrix. The procedure cosists of usig elemetry row trsformtios to trsform the give mtrix ito trigulr mtrix i the ove exmple ito upper trigulr mtrix), tig ito ccout the effect of ech trsformtio, the filly evlutig the determit of the resultig trigulr mtrix y multiplyig the etries log the mi digol.. PROBLEMS. Use elemetry row opertios to evlute the determits of the followig mtrices. ) ) c) d) e) f) g). PROPERTIES OF DETERMINANTS Let A e squre mtrix. Let Â e the mtrix resultig from performig oe or more elemetry row opertio o A. Sice the effect of performig elemetry row opertio o the vlue of the determit is either to reverse the sig or multiply the vlue of the determit y ozero umer, d sice the elemetry row opertios rows & iterchged row hs fctor multiples of row dded to rows & product of etries o mi digol

9 re ivertile opertios; therefore det Â) if d oly if deta) d similrly det Â) if d oly if deta). Let sequece of elemetry row opertios e performed o the mtrix A so s to reduce A to its reduced row-echelo form R. Now A is ivertile if d oly if R I. But det R) det I) if d oly if det A). We therefore coclude tht A is ivertile if d oly if deta) d stte this result i the form of theorem. Theorem The squre mtrix A is ivertile if d oly if det A). Exmple det so is ivertile mtrix. det so is ot ivertile. A direct cosequece of the ove theorem is the followig result. Theorem Let A e squre mtrix. The the lier system Ax hs uique solutio for every if d oly if det A). Proof Suppose det A), the A is ivertile. The A is solutio to Ax sice AA ). To show tht this is the oly solutio to Ax, suppose tht x is lso solutio to Ax. The Ax so A A x ) A d hece x A. This shows tht A is the uique solutio to Ax. O the other hd, if Ax hs uique solutio, the whe solvig this system y mtrix methods the coefficiet mtrix is reduced to the idetity mtrix I d so A is ivertile d hece deta). Theorem Let A e mtrix d let E e elemetry mtrix. The detea) dete) deta). Proof The proof cosists i showig tht the result is true for ech oe of the three types of elemetry mtrices. Let E e the elemetry mtrix otied from I y iterchgig two rows of I. The EA is the mtrix resultig from iterchgig the correspodig two rows of A. The detea) deta) ) deta) dete) deta) sice dete). Let E e the elemetry mtrix otied from I y multiplyig the etries of some row of I y ozero sclr. The EA is the mtrix resultig from multiplyig the etries of row of A y. The detea) deta) dete) deta) sice dete). Let E e the elemetry mtrix otied from I y ddig multiple of oe row of I to other row of I. The EA is the result of ddig multiple of row of A to other row of A. The detea) deta) ) deta) dete) deta) sice dete). The result i the ove theorem c e geerlized to y two is omitted ut stted i the followig theorem. mtrices. The proof

10 Theorem If A d B re squre mtrices of the sme size, the detab) deta) detb). Exmple Let A, d let B, the AB. deta) detb) d det AB) deta) detb) )) detab). Theorem If A is ivertile mtrix, the deta ) /deta) Proof A - A I deta - A) deti) deta - )deta) deta - ) / deta). Theorem If A is squre mtrix, the deta T ) deta). Proof A cofctor expsio log the first row of A T gives the sme terms s cofctor expsio log the first colum of A. Exmple Let A, the A T. deta) )) )). det A T ) )) )). So det A) deta T ). Theorem Let A d B e squre mtrices with AB I. The BA I. Proof We first show tht there exists mtrix such tht A I d the show tht i fct B. Sice AB I d sice deti), therefore deta). But deta T ) deta), so deta T ) d hece A T is ivertile. Let D deote the iverse of A T ; so D A T ). T T T T T T T T T The A D I A D) I D A ) I D A I so D T. We ow show B s follows. I AB) A)B IB B. Whe computig the iverse of mtrix A oe should verify the correctess of the computtio y demostrtig tht oth the products AA d A A equl I. The precedig theorem proves tht i fct tht it is sufficiet to show tht oly oe of these two products eeds to e show equl to I.

11 . PROBLEMS. Determie whether the mtrix is ivertile or ot y clcultig its determit. ) ) c) d) e). Use determits to show tht the followig systems of lier equtios hve uique solutios. x y ) ) x y z x y z x y c) x y z d) x y z x y x y x y z x y z. Let A d B e mtrices with deta) d detb). Fid the followig. ) detab) ) deta ) c) detab ) d) detab) e) deta T B). Let A d B e mtrices with deta) d detb). Fid the followig. ) detab) ) deta ) c) detab ) d) detab) e) deta T B). THE ADJOINT MATRIX Recll the th row cofctor expsio of mtrix A for ws defied to e deta) ) deta ) ) deta ) ) deta ) where j the qutity ) deta j ) is clled the j th cofctor of A. To simplify our ottio we will deote this qutity y the symol j. Thus the th row cofctor expsio for deta) c e writte more simply s deta). Suppose tht i this expressio we replce the th row etries,, y the j th row etries j, j, j to get j j j. Such expressio would rise if the etries of the th row of A were replced y the etries of the j th row of A d cofctor expsio log this ew th row were doe. But the vlue of this determit would e sice the mtrix hs two ideticl rows rows d j re sme). We hve thus estlished the followig theorem. Theorem If A is squre mtrix, the deta) if i i j if i j j I similr fshio, we c deduce the followig result for colum cofctor expsio.

12 If cofctor expsio log colum is used the deta) if i j i j. if i j ofctor Mtrix Let A e squre mtrix. The cofctor mtrix of A, deoted cofa) is the mtrix otied from A y replcig every etry of A y its cofctor. Exmple If A, the cofa). Adjoit Mtrix If A is squre mtrix, the djoit of A, deoted dja) is the trspose of the cofctor mtrix; tht is dja) [cofa)] T. Exmple otiuig with the previous exmple, A, cofa) d so dja) [cofa)] T osider ow the product A dja) But deta) )det )det )det )) )) )). We see tht the product This exmple suggests the followig theorem. A dja) is digol mtrix with the digol etries deta).

13 Theorem If A is squre mtrix, the I deta) dja) A Proof dja) A I deta) deta) deta) deta) deta) deta) deta) Theorem If deta), the A is ivertile d A dja). deta) Proof From the previous theorem we hve I deta) dja) A. Sice deta) we c divide y deta) to get I dja) deta) A so A dja). deta) Exmple Use the precedig theorem to fid A for the mtrix. A Solutio This is the sme mtrix used i the previous exmple where we foud dja) d deta). Usig A dja) deta) we get A / / / / / / / / /.

14 . PROBLEMS. For ech of the followig mtrices fid its djoit, the use the djoit d the vlue of the determit) to fid the iverse of the mtrix. ) ) c) d) e) f). Let A e mtrix with deta). Show tht det[dja)] [deta)].. Let A e mtrix with deta). Fid det[dja)].. Let A e mtrix with deta). Fid det[dja)].. Let A e mtrix with deta). Fid det[dja)].. RAMER'S RULE rmer's rule provides formul for solvig system of lier equtios i vriles whe the system hs uique solutio. Theorem rmer's Rule) Let Ax e system of lier equtios i vriles with deta). Let A e the mtrix otied from A y replcig the th colum of A y the colum vector. The the system hs the uique solutio deta x ),,,,. deta) Proof Sice deta), A is ivertile d the system Ax hs the uique solutio x A. Therefore x dja) deta) deta)

15 deta). Thus.,,, for deta) x Now the umertor of x cosists of the th colum cofctors of A multiplied y the correspodig etries of. We get the sme result if we use th colum cofctor expsio of A so deta) ) deta x for,., Exmple Use rmer's rule to solve the followig system of lier equtios. z y x z y x z y x Solutio det det x det det y det det z. PROBLEMS. Use rmer s rule to solve the followig systems of lier equtios. ) x y ) x y c) x y x y x y x y d) x y z e) x y z f) x y z x y z x y z x y z x y z x y x y z

16 . APPLIATIONS USING THE DETERMINANT I this sectio two pplictios usig the determit will e cosidered. The first is coveiet method for evlutig cross product. The secod pplictio is fidig the volume of prllelepiped. ross Product Recll if u u, u, u) d v v, v, v ), the u v uv uv, uv uv, uv uv ) uv uv ) i uv uv ) j uv uv ) where i,, ), j,, ) d,, ). A coveiet method for rrivig t the cross product u v is to use the mtrix form i j u u u d do first row cofctor expsio s you would do to evlute v v v determit. Exmple Let u,, ) d let v,, ). Fid u v. Solutio i j u v i j i j,,) Volume of Prllelepiped Recll tht if prllelepiped hs the vectors u, v d w s edges, the the volume of the prllelepiped is w u v). A stright forwrd w w w u u u clcultio shows tht w u v) det u u u det v v v v v v w w w Hece the volume of the prllelepiped c e foud y tig the solute vlue of either of the ove determits. Exmple Fid the volume of the prllelepiped hvig the vectors,, ),,, ) d,, ) s edges. Solutio Volume det.

17 . PROBLEMS. lculte the followig cross products. ),,),, ) ),,),,) c),,),, ) d),,),, ). Fid the volume of the prllelepiped hvig the followig vectors s edges. ),, ),,, ),,, ) ),, ),,, ),,, ) c),, ),,, ),,, ) d),, ),,, ),,, )

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