Chapter 2. Determinants


 Nickolas Porter
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1 Chpter Determinnts
2 The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is denoted by the symbol A or det(a). With this nottion, the formul for A is det( A) d c b We wnt to obtin the nlogous formul to squre mtrices of higher order.
3 Cofctor Expnsion: Crmer's Rule Minors nd Cofctors Definition: If A is squre mtrix, then the minor of entry ij is denoted by M ij nd is defined to be the determinnt of the submtrix tht remins fter the ith row nd jth column re deleted from A. The number ( ) i+j M ij is denoted by C ij nd is clled the cofctor of entry ij. Exmple: Find the minors nd cofctors of: The minor of entry is The cofctor of is C = ( ) + M = M = Similrly he minor entry is The cofctor of is C = ( ) + M = M = A M M
4 The cofctor nd minor of n element differ only in sign. To determine the sign relting C ij nd M ij, we my use the checkerbord rry For exmple, C = M, C = M, C = M, C = M nd so on. Cofctor Expnsion: Crmer's Rule 4
5 Cofctor Expnsion: Crmer's Rule Cofctor Expnsions The definition of determinnt in terms of minors nd cofctors is det(a)= M  M + M = C + C + C Where M This method of evluting det(a) is clled cofctor expnsion long the first row of A. Exmple: Evlute det(a) by cofctor expnsion long the first row of A. Solution:, M A 5 4 4, M 0 5
6 Cofctor Expnsion: Crmer's Rule If A is mtrix then its determinnt is: A det(a)= M  M + M = (  ) (  )+ (  ) = = C + C + C = C + C + C = C + C + C = C + C + C = C + C + C = C + C + C These equtions re clled the cofctor expnsions of
7 Theorem: The determinnt of n nn mtrix A cn be computed by multiplying the entries in ny row (or column) by their cofctors nd dding the resulting products; tht is, for ech in nd jn, Cofctor expnsion nd row or column opertions cn sometimes be used in combintion to provide n effective method for evluting determinnt. Exmple: Evlute det(a) by cofctor expnsion long the first row nd det(b) by cofctor expnsion long the third column ) ( ) det( ) ( ) det( column j C C C A row i C C C A th nj nj j j j j th in in i i i i A Cofctor Expnsion: Crmer's Rule B 7
8 Cofctor Expnsion: Crmer's Rule Smrt Choice of Row or Column 8
9 Cofctor Expnsion: Crmer's Rule Entries nd Cofctors from Different Rows Definition: If A is ny nn mtrix nd C ij is the cofctors of ij, then the mtrix C C C n C C C n is clled the mtrix of cofctors from A. The trnspose of this mtrix is clled the djoint of A nd is denoted by dj(a). C C C n n nn 9
10 Cofctor Expnsion: Crmer's Rule Exmple: Find the djoint of A: A 4 0 0
11 Cofctor Expnsion: Crmer's Rule Using the Adjoint to Find n Inverse Mtrix Theorem: If A is n invertible mtrix, then A det( A) dj( A) Exmple: Find the inverse of A in lst exmple (using the djoint concept). Solution: det(a) =4
12 Cofctor Expnsion: Crmer's Rule Tringulr Mtrices Theorem: If A is n nn tringulr mtrix (upper tringulr, lower tringulr, or digonl), then det(a) is the product of the entries on the min digonl of the mtrix; tht is: det(a)=... nn Exmple: The determinnt of Exercise: If A = [ ij ] is n invertible tringulr mtrix, the djoint formul for A is the successive digonl entries of A re, ,, ()( )()(9)(4) 9 nn
13 Cofctor Expnsion: Crmer's Rule Crmer's Rule Theorem: If Ax=b is system of n liner equtions in n unknowns such tht det(a)0, then the system hs unique solution. This solutions is x A A A A det det, x det det,, x n det det An A where A j is the mtrix obtined by replcing the entries in the jth column of A by the entries in the mtrix b b b b n
14 4 0 A A 8 0 A A ) det( det( ) A A x ) det( det( ) A A x ) det( det( ) A A x Cofctor Expnsion: Crmer's Rule Exmple: Use Crmer's rule to solve Solution: x x x x x x x x 8 0 b 4
15 Evluting Determinnts by Row Reduction Bsic Theorm: Theorem: Let A be squre mtrix, then () If A hs row of zeros or column of zeros, then det(a)=0. (b) det(a) = det(a T ). Theorem: Let A be n n n mtrix, then ()If B is the mtrix tht results when single row or single column of A is multiplied by sclr k, then det(b)=k det(a). (b)if B is the mtrix tht results when two rows or two columns of A re interchnged, then det(b) = det(a). (c) If B is the mtrix tht results when multiple of one row of A is dded to nother row or when multiple of one column is dded to nother column, then det(b) = det(a). 5
16 Exmple: A
17 Evluting Determinnts by Row Reduction Elementry Mtrices Theorem: Let E be n n n elementry mtrix, then ()If E results from multiplying row of I n by k, then det(e) = k. (b)if E results from interchnging two rows of I n, then det(e) = . (c) If E results from dding multiple of one row of of I n to nother, then det(e)=. Exmple: Determinnts of Elementry Mtrices 7
18 Evluting Determinnts by Row Reduction Mtrices with Proportionl Rows or Columns Theorem: If A is squre mtrix with two proportionl rows or two proportionl columns, then det(a) = 0. Exmple: Introducing Zero Rows 8
19 Evluting Determinnts by Row Reduction Evluting Determinnts by Row Reduction Ide: to reduce the given mtrix to upper tringulr form by elementry row opertions, then compute the determinnt of the upper tringulr mtrix, then relte tht determinnt to tht of the originl mtrix. The method is well suited for computer evlution since it is systemtic nd esily progrmmed. 9
20 Exmple: Using Row Reduction to Evlute Determinnt Evlute det(a) where Solution: We will reduce A to rowechelon form (which is upper tringulr) nd pply previous theorem A 0
21 Exmple: Using Column Opertions to Evlute Determinnt
22 Exmple: Row Opertions nd Cofctor Expnsion Evlute det(a) where Solution: By dding suitble multiples of the second row to the remining rows, we obtin
23 Properties of the Determinnt Function Bsic Properties of Determinnts Let A nd B be nn mtrices nd k be ny sclr. We hve det(ka)=k n det(a). det(a+b) is usully not equl to det(a)+det(b). Exmple: Exmple: A, 5 B A B 4 8 A 5 A 5 det( A), det( B) 8, det( A B) det( A), det(a) 9 Thus det A + det(b) det(a + B) Thus det(a) det( A)
24 Properties of the Determinnt Function Interesting exmple: A nd B b b Theorem: Let A, B, nd C be nn mtrices tht differ only in single row, sy the rth, nd ssume tht the rth row of C cn be obtined by dding corresponding entries in the rth rows of A nd B. Then det(c) = det(a) + det(b). The sme result holds for columns. 4
25 Properties of the Determinnt Function Exmple: Determinnt of Mtrix Product Lemm: If B is n nn mtrix nd E is n nn elementry mtrix, then det(eb)=det(e)det(b). It follows tht Det(E E...E r B)=det(E )det(e )...det(e r )det(b) 5
26 Properties of the Determinnt Function Determinnt Test for Invertibility Theorem: A squre mtrix A is invertible if nd only if det(a)0. A squre mtrix with two proportionl rows or columns is not invertible. Exmple: Theorem: If A nd B re squre mtrices of the sme size, then det(ab)=det(a)det(b).
27 Properties of the Determinnt Function Exmple: Verifying det(ab)=det(a)det(b) A, B 5, 8 AB 7 4 det( A), det(b) , det(ab)  Theorem: If A is invertible, then det( A ) det( A) 7
28 Properties of the Determinnt Function Liner Systems of the Form Ax = λx Mny pplictions of liner lgebr re concerned with systems of n liner equtions in n unknowns tht re expressed in the form Ax = λx, where λ is sclr. Such systems re relly homogeneous liner systems in disguise, since it cn be rewritten s Ax λx = 0, or, by inserting n identity mtrix nd fctoring, s λi A x = 0. Such vlue of λ is clled chrcteristic vlue or n eigenvlue of A. If λ is n eigenvlue of, then the nontrivil solutions of λi A x = 0 re clled the eigenvectors of A corresponding to λ. It follows tht the system λi A x = 0 hs nontrivil solution if nd only if det λi A = 0. This is clled the chrcteristic eqution of A; the eigenvlues of A cn be found by solving this eqution for λ. 8
29 Properties of the Determinnt Function Exmple: Find the eigenvlues nd corresponding eigenvectors of the system Solution: The liner system cn be written in mtrix form s which is of form λi A x = 0 with This system cn be rewritten s The chrcteristic eqution of A is The fctored form of this eqution is λ + λ 5 = 0, so the eigenvlues of A re λ = nd λ = 5 (see next slide) 9
30 Properties of the Determinnt Function Exmple: The eigenvlues nd eigenvectors (cont.) By definition, is n eigenvector of A if nd only if X is nontrivil solution of λi A x = 0; tht is: 0
31 Properties of the Determinnt Function Summry Theorem: Equivlent Sttements If A is n nn mtrix, then the following re equivlent. ()A is invertible. (b)ax=0 hs only the trivil solution. (c) The reduced rowechelon form of A is I n. (d)a is expressible s product of elementry mtrices. (e)ax=b is consistent for every n mtrix b. (f) Ax=b hs exctly one solution for every n mtrix b. (g)det(a)0.
32 Combintoril Approch to Determinnts
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40 Combintoril Approch to Determinnts signed elementry product from A: n elementry product multiplied by + or. We use + if (j,j,...,j n ) is n even permuttion nd the if (j,j,...,j n ) is n odd permuttion. Exmple: List ll signed elementry products from the mtrices 40
41 Combintoril Approch to Determinnts Definition: Let A be squre mtrix. The determinnt function is denoted by det, nd we define det(a), determinnt of A, to be the sum of ll signed elementry products from A. Exmple: Find the determinnts of mtrices ( ) A det(a)  ( b) B det( B) 4
42 Evlute the determinnts by product of rrows (only for nd mtrices) Determinnt of mtrix Determinnt of mtrix Exmple: Evlute the determinnts of A 4 nd B
43 Combintoril Approch to Determinnts Nottion nd Terminology The symbol A is n lterntive nottion for det(a). Although the determinnt of mtrix is number, it is common to use the term determinnt to refer to the mtrix whose determinnt is being computed. The determinnt of A is often written symboliclly s det( A) Evluting determinnts directly from the definition leds to computtionl difficulties. j j nj n 4
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