4. Determinants. det : { square matrices } F less important in mordern & practical applications but in theory

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1 4 Determiats det : { square matrices } F less importat i morder & practical applicatios but i theory ew formula for solvig LES ew formula for iverse of a matrix test if a matrix is regular calculate area ad orietatio or parallelogram ad parallelepiped 4 Determiats of order 2 Notatio: a a a a A M order of determiat a a a a a b Defiitio 4 A M 2 2 F) A ad bc c d Example 42 A B A+B deta detb deta+b) 4 determiat is ot liear but it is liear i row additio Theorem 43 Let u,v,w F 2, k F u+kv u v det det +kdet w w w put 2 row vectors above / below each other to get a 2 2 matrix det w u+kv det w u put 2 col vectors beside each other) +kdet w v Proof: exercise first applicatio of determiat: detect ivertibility Theorem 44 A A ij ) M 2 2 is ivertible 0 The A A22 A 2 A 2 A

2 2 Proof Assume 0 The A22 A 2 A 2 A A22 A 2 A A 2 0 A A 2 A A 2 A 22 0 similarly A Now assume A is ivertible rka) 2 rk 2 rk 2 A A 2 A A 2 A 0 A A 2 A A 2 A 22 0 A 22 A 2A 2 A }{{ } 0 0 A 2 0 A A 2 rk {}}{ 0 A 2 A A 22 A 2 A 2 A 22 Area of parallelogram A det u v) sg det u v is orietatio θ > 0 positive θ < 0 egative 42 Determiats of order

3 Defiitio 45 σ : N {}}{ {,,} permutatioσ) σ)) N {,,} bijective is called S : {σ : N N } symmetric group ) 3542) S is a group with multiplicatio give by compositio στ : σ τ 432) 234) 432) 234) 342) S! 2 sig: I ), ) σ sgσ) : ) # sig of σ remark: o!!! example 2 ) sg ) 0 sg 2 2 ) 2 sg ) ) ) 7 Whe sgσ), we call σ eve or positive), whe sgσ), we call σ odd or egative) sg : S {,+} is multiplicative: sgτσ) sgτ)sgσ) Defiitio 46 A A ij ) A M square matrix!) determiat of A ) σ σ S A A 2 Example 47 2 A A 2 A 22 σ σ 2 j A jσj) ) σ 2 2 A jσ j) + ) σ 2 j j A jσ2 j) ) }{{} A A 22 σ ) σ 2) + ) }{{} A 2 A 2 σ 2 ) σ 2 2) A A 22 A 2 A 2

4 4 Example 48 3 A A 2 A 3 A 2 A 22 A 23 A 3 A 32 A 33 { + S } ) A A 22 A 33 + ) A A 23 A 32 + ) A 2 A 2 A 33 + ) A 3 A 2 A 32 + ) A 2 A 23 A 3 + ) A 3 A 22 A 3 A A 22 A 33 +A 2 A 23 A 3 +A 3 A 2 A 32 A A 23 A 32 A 2 A 2 A 33 A 3 A 22 A 3 gives volume of parallelepiped 평행육면체 ) Why am I doig this? Theorem 49 det is liear i each row/colum det a a i a i +k b a i+ a det a a i a i a i+ a + k det a a i b a i+ a 2)  A à det a ai a i + k b ai+ a det a a i a +k det a ai b ai+ a Proof detâ σ σ σ σ i ) σ j i ) σ ) σ ) σ j j j i j j i  jσj) deta+k detã)  jσj)  jσj) ji ji+  jσj) aiσi) +kb σi) )  jσj) a iσi) +k σ A jσj) A iσi) +k σ ji+ ) σ ) σ  jσj) j j i j j i Corollary 40 If A has a zero row or colum, the 0  jσj) b σi) à jσj) Ãσi)

5 5 Theorem 4 det a a 2 det a 2 a & more geerally for ay 2 rows ad colums }{{}}{{} A Â 2 2 Proof Letτ,2) Thesgτ) i i i 3,, ad sgστ) sgσ) Thus multiplicatio by τ gives a bijective map {odd/eve permutatios} σ ˆσ : σ τ {eve/odd permutatios} The detâ σ σ σ σ ˆσ ˆσ ) σ a jσj) a σ2) a 2σ) j3 ) σ a jˆσj) a ˆσ) a 2ˆσ2) j3 )ˆσ j a jˆσj) )ˆσ a jˆσj) deta Corollary 42 If A has two equal rows or colums, the 0 for charf) 2; otherwise exercise) j Corollary 43 det is preserved uder row & colum operatios of type III deti ) add to a row a multiple of aother row) det chages sig uder uder row & colum operatios of type I exchage 2 rows or 2 colums) det multiplies uder row & colum operatios of type II multiply row) det ) 0 oly if σ Id σ so ca calculate determiat usig row & colum operatios:

6 complexity is 3 3 Aother less practical) way : developmet complexity! Ã ij a ij delete i-th row ad j-th colum i,j)-mior of A row develop det A colum develop det A ) k+j a kj detãkj j ) k+j a kj detãkj k for every k,, for every j,, Theorem 44 A ivertible deta 0 Proof A ivertible A Id by row ad col op det 0 do ot chage det 0 A o-iv A zero row/col matrix deta 0 동상 ) 43 More) properties of determiat Theorem 45 deta T ) Proof A Id A T Id trasposed operatios chage determiat i same way Theorem 46 3) deta B) detb) If deta B) 0 0 or detb) 0 A B ot ivert 4) A ot ivert or B ot ivert

7 4) A& B ivert AB ivert if A ot ivert if B ot ivert A square mx kera) {0} kerba) {0} B square mx B ot surj BA ot surj 7 so assume both had sides are ot zero i 3) ow A is ivertible A product of elemetary matrices eough to prove 3) for A elemetary matrix A A 0 λ 0 det det λ AB B with row i & j exchaged Thus detab) detb) detb) 3) ok AB B with row i multiplied by λ Thus detab) λdetb) detb) 3) ok A α +α det detid ) AB B with row i+ α row j Thus detab) detb) detb) 3) ok A formula for the matrix iverse A ) ij detãji) ) i+j 2 3 Example 47 A had before deta 3 check by developmet rule: deta 8 0) 2+3 3) 3 ok à 8 8 à à detã 8 detã2 0 detã3 3

8 8 A A deta detã detã2 detã3 detã 2 detã 22 detã detã3 detã23 detã33 8/ 3 8/ 3 0/ 3 3/ This is ot practical Gauss elimiatio better), but has some theoretical applicatios eg A M Z) deta ± A M Z) a cosequece of this formula is detã detã detã detã detã detã Theorem 48 Cramer s rule) The uique) solutio x A b of Ax b for A square matrix ivertible x x,,x ) T x k detm k deta M k replace colum k of A by b Example 49 x +2x 2 +3x 3 2 x +x 3 3 x +x 2 x A 0 b 3

9 x detm ) x 2 detm 2) x 3 detm 3) 2 det det det x 5 2,, 2 ) 9 agai, this is ot practical for large as Gaussia elimiatio!) but it saysagai: if deta ±, A, b have iteger etries, the x is iteger solutio