Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)

Transcription

1 5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper Triagular Matrix 5.6. Computatio of ay Determiat by the Gauss-Jorda Process 5.7. Uiqueess of the Determiat Fuctio 5.8. Exercises 5.9. Multiliearity of Determiats 5.0. Applicatios of Multiliearity 5.. The Product Formula for Determiats 5.2. The Determiat of the Iverse of a Nosigular Matrix 5.3. Determiats ad Idepedece of Vectors 5.4. The Determiat of a Bloc-Diagoal Matrix 5.5. Exercises 5.6. Expasio Formulas by Cofactors 5.7. The Cofactor Matrix 5.8. Cramer s Rule 5.9. Expasio Formulas by Miors Exercises 5.2. Existece of the Determiat Fuctio Miscellaeous Exercises o Determiats Summary

2 5.. Itroductio Determiats of order 2 ad 3 were defied i Chapter 2 by the formulae a a a 2 a 2 22 a a a a (5.) a a a a a a a a a a a a a a a (5.2) a32 a33 a3 a33 a3 a a a a The similarity i form betwee determiats ad matrices suggests a close relatioship betwee them. Ideed, we shall treat the determiat as a fuctio that maps a square matrix to a umber, i.e., det : M, R or C by A det A ai where M, is the space of all matrices ad the fuctio value det A ai is to be evaluated by some geeralized versio of eqs(5.-2). There are may differet, but equivalet, ways to carry out the geeralizatio. Oe way that emphasizes the ati-symmetry of the determiat fuctio is to defie it as, det A det ai i, a a a 2 a a a a a a 2 P P 2 a a a P 2P P where P is a permutatio of itegers, i.e., P P P P,2,,, 2,, ad P if P is a eve odd permutatio. I geeral, there are! terms i the sum. For example, the case 3 has 6 terms: P P 2 P 3 () P

3 so that a 3,3 aa22a33 aa23a32 aa 2a33 aa 23a3 aa 2a32 aa 22a3 det i i, as ca be checed agaist (5.2) with the help of (5.). O the other had, there is also a geometrical iterpretatio of a determiat as a volume i E. This will play sigificat roles i may area of mathematics, e.g., multi- dimesioal itegrals, ad exterior derivatives. I the followig sectios, we shall explore some elemetary features of this aspect of the determiat fuctio.

4 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios It was proved i Chapter 2 that the scalar triple product of 3 vectors,, A a, a, a A a, a, a A a a a 2 3 ca be writte as A A A2 A3 det A 2 A 3 det a a a 2 3 a a a a a a which is o-zero iff the row vectors are idepedet. Furthermore, it is easily show geometrically that A A A is equal to the volume of the parallelepiped 2 3 whose sides are A A 2 ad A 3 [see Fig.5.]. If the vectors are depedet, they are coplaar so that the triple product, ad hece the volume, vaishes. Later o, we shall defie the geeral determiat fuctio d as a -D volume. The 3-D case thus serves as a prototype from which the basic characteristics of such a volume fuctio ca be deduced. [ Note: it is more rigorous, as well as simpler, to start with the 2-D case ad deal with a area fuctio. Nevertheless, we shall follow Apostol s approach here]. Cosider the the determiat fuctio,, d A A A A A A As a volume fuctio, it satisfies the followig easily verified geometrical properties:. If a row (side) is multiplied by a factor t, so is the determiat (volume). 2. If a row (side) is replaced by the sum of itself ad aother row (vector sum of it ad aother side), the determiat (volume) is uchaged. 3. The determiat (volume) of the 3 uit coordiate vectors, which forms the uit cube, is. These will serve as the bases for the axiomatic defiitio of the determiat fuctio to be described i the ext sectio.

5 5.3. A Set of Axioms for a Determiat Fuctio Defiitio: Determiat Fuctio Theorem Theorem Theorem 5.3

6 5.3.. Defiitio: Determiat Fuctio Let A a i be a matrix with ai R or C. We deote its ith row by A a, a,, a so that i i i2 i A A A The determiat fuctio of order is defied as R C by A A d A A d : M, or such that the followig axioms are satisfied: Axiom. Homogeeity I Each Row:,,,, i i det,, d ta td A where t R or C. Axiom 2. Axiom 3. Ivariace Uder Row Additio: Scale:, i,,,, i,,, d A A A d A A det,, I d I I where I is the th uit coordiate vector.

7 Theorem 5.. If some row of A is the zero vector, the det A 0. Proof Let A O. Sice ta O A, we have, with t,,,,, d A d A However, by axiom, we have Hece, QED.,,,, d A d A d, A, d, A, 0

8 Theorem 5.2., i,,,, i,,, d A ta A d A A where i Proof ad t is a scalar. The case t 0 is trivial so that we shall assume t 0. By axiom, we have, i,,,, i,,, d A ta td A A By axiom 2, we have, i,,,, i,,, d A ta d A ta ta td, A ta,, A, [axiom used] i Comparig with the st equatio fiishes the proof.

9 Theorem 5.3. (a) d, Ai,, A, d, A,, Ai, for all i. (b) d, A,, A, 0 i i Proof of (a) By repeated use of theorem 5.2, we have, i,,,, i,,, d A A d A A A, i,, i, d, Ai A,, Ai, d A A A A A, i i,, i, d, A,, Ai, d A A A A d, A,, A, [axiom used] i Proof of (b) Usig (a), we have d, A,, A, d, A,, A, 0 i i i i

10 5.4. The Determiat of a Diagoal Matrix Defiitio: Diagoal Matrix A matrix A a i I other words, it has the form a a22 0 A 0 0 a ad ca be deoted A diag a, a22,, a Theorem 5.4. If A diag a a a is called a diagoal matrix if a 0 wheever i., 22,,, the i det A a a a 22 a (5.4) i ii Proof Usig I to deote the ith uit coordiate vector, we ca write A as A ai, a22i2,, ai Applyig axiom repeatedly, we have a d I a I a I det A d ai, a22i2,, ai 22 2,,, a a d I I a I a a a d I, I,, I det,,, 22 2 a a a I (5.5) a a a [ axiom 3 used ] 22

11 5.5. The Determiat of a Upper Triagular Matrix The Gauss-Jorda elimiatio process is also oe of the best methods for calculatig determiats. For coveiece, we summarize the effects of the elemetary row operatios [ 4.8] o the value of the determiat. Operatio Effect Proof Iterchage 2 rows deta deta Theorem 5.3 Multiply a row by scalar t deta t deta Axiom Add to oe row a scalar multiple of aother deta deta Theorem 5.3 We shall show that the last operatio aloe is sufficiet to trasform a upper triagular matrix ito a diagoal oe. Defiitio: Upper Triagular Matrix A upper triagular matrix is ay matrix of the form U u u u 0 u u 0 0 u (5.6) so that all elemets below the diagoal vaish, i.e., ui 0 i. Theorem 5.5. For a upper diagoal matrix U, Proof detu u u u 22 u (5.7) i ii It is easy to see that if we apply the bacward substitutio process of the Gauss-Jorda method [see 4.8] to (5.6), we ca trasform U to the equivalet form U diag u, u22,, u provided uii 0 i. Applyig Theorem 5.4 the gives (5.7). If uii 0 for some i, the process breas dow because row i becomes a zero vector. However, by Theorem 5., we have detu 0 so that (5.7) agai holds. QED.

12 5.6. Computatio of ay Determiat by the Gauss-Jorda Process The forward elimiatio phase of the Gauss-Jorda process ca be used to trasform a geeral square matrix A to a upper triagular form U. 5.5, it is clear that p detu c c det A q With referece to the table of where p is the umber of times rows were iterchaged ad c,, cq are the q ozero scalars used to multiply rows durig the elimiatio process. Hece, det A p detu c c q p c c q u u det I (5.8a) Note that Axiom 3 eters these cosideratios oly to fix the value of det I. Hece, axioms ad 2 aloe is eough to give det A c A det I (5.8) where c A is a scalar that ca be calculated as i (5.8a).

13 5.7. Uiqueess of the Determiat Fuctio Eq(5.8) showed that axioms ad 2 aloe ca be used to defie a determiat fuctio. This raises the questio of uiqueess of such fuctios. Theorem 5.6. Uiqueess Theorem For Determiats Let d be a fuctio satisfyig all 3 axioms for a determiat fuctio of order. is aother fuctio satisfyig axioms ad 2, the for every matrix A, we have f A d A f I (5.9) If f Hece, if f also satisfies axioms 3, the f A d A so that d is uique. Proof Cosider g A f A d A f I (a) Sice both f ad d satisfy axioms ad 2, so does g. However, usig d I, we have g I f I d I f I 0 so that g does ot satisfy axiom 3. Thus, eq(5.8) becomes c A g I 0 g A Puttig this ito eq(a), we recover eq(5.9). QED.

14 5.9. Multiliearity of Determiats Theorem 5.7. Additivity I Each Row. Give a matrix A A A,, ad ay -D vector V. Let B ad C be matrices that differ from A oly i the th row, amely, The, or,,, B, V, C, A V, A A det C det A det B (5.0),,,,,, d A V d A d V (5.0a) which is ow as the additivity i the th row. Proof Let f A det C det B d, A V, d, V, (a) where the right had side of the last equality is assumed to be idepedet of V. shall prove the theorem by showig that f A satisfies all 3 axioms for a We determiat fuctio so that by the uiqueess theorem, f A det A. For operatios ot ivolvig A, the proof that f A satisfies axioms ad 2 follows trivially from the fact that detc ad detb are determiat fuctios. operatios ivolvig A, the proof for axiom is as follows,,,,,, f ca d ca V d V For V V cd, A, cd,, c c V V c d, A, d,, c c c f, A, where the last equality made use of the assumptio that detc detb is idepedet of

15 V. For axiom 2, we have,,,,,,,,,,,, f A A A d A A V A d V A,,,,,,,, d A V A d V A f, A,, A, ad,,,,,,,,,,,, f A A A d A V A A d V A A Applyig the theorem itself to the right had side gives,,,,,,,,,,,, f A A A d A V A d A V A where we ve used,,,,,,,, d V A d V A f, A,, A,,,,,,,,,,,,, d A V A d A V A A d V A Fially, we eed to prove f satisfies axiom 3, amely, f I. Settig A I i eq(a), we have,,,,,,,, f I d I I V I d I V I Let V v v,,. If, both C ad B are upper triagular. By Theorem 5.5, we have det C v ad det B v so that (b) f I v v (c) ad the theorem holds. For 2, we have

16 v v2 v3 v v v2 v3 v C B To mae each of these upper triagular, we eed oly add v st row to the 2 d row. The operatio does ot affect the diagoal elemets so that we agai recover eq(b) ad hece eq(c). Furthermore, a little reflectio will show that the same holds for all so that the theorem is proved. Multiliearity By repeated applicatios of Theorem 5.7 ad Axiom to the same row, we ca geeralize (5.0a) to p p d, tu i i, tid, Ui, i i which expresses the liearity i each row. Sice this applies to every row, the determiat fuctio is said to be multiliear fuctio i its rows.

17 5.0. Applicatios of Multiliearity Theorem Theorem 5.9.

18 5.0.. Theorem 5.8. If the rows of a matrix A are depedet, the d A 0. Proof Sice the rows A,, A are depedet, the equatio i c A i i O ca be satisfied with some ozero ci A ti Ai where ti 0 c i Thus, by the liearity i the th row, we have d A d i i i c s. Let c 0, the we ca write,,, t A, t d, A, Note that every,, i i i d A i i the sum vaishes because the ith ad th rows are idetical [see Theorem 5.3(b)]. QED. i

19 Theorem 5.9. (a) For matrices of order 2, there is ad oly determiat fuctio, amely, a a a a det a a a a a a a a (5.2a) (b) For matrices of order 2, there is ad oly determiat fuctio, amely a a a a a a det a a a a a a a a a a a a a a a a a a a a a (5.2) a32 a33 a3 a33 a3 a32 Proof of (a) The easiest way is to show that the formula satisfies all 3 axioms for a determiat fuctio ad the ivoe the uiqueess theorem. However, it is more istructive to derive them directly as follows. Give a A a a 2 a 2 22, we have A a a, 2 a a2 i A a, a a2i a22 where i,0 ad 0, are the uit coordiate vectors. Usig the liearity i the st row, we have, i, a d i, A a d, A d A A d a a A Usig the liearity i the 2 d row, we have i, i, i a d i, i a d i, a d i d A d a a ,,, i a d, i a d, a d i d A d a a , where we have used d i, i d, 0. Usig d, i d,, i,, i a a a a d i d A A a a d a a d , i, we have

20 Usig d 0 i, d I the completes the proof. Proof of (b) This is left as a exercise. 0 Geeral Formula Let A be a matrix. Its th row ca be writte as A a,, a a I where 0,,,,0 I is the th Cartesia basis vector. Thus,,,,, d A d A A A d a I,,,, a I a I a a a d I,, I,, I Now, d I,, I,, I 0 if,,,, is a eve odd ot permutatio of,,,, Therefore, d A is a siged sum of product terms of the form a a a, where the set of idices,,,, is usually deoted by d A a a a P P P P P where P stads for permutatio. is a permutatio of,,,,. This

21 5.. The Product Formula for Determiats The product of a m matrix A a i ad a p matrix B b i is defied as the m p matrix C c i with compoets [see 4.5] c a b (5.3) i i Lemma 5.0. Give a m matrix A a i ad a p matrix B b i AB i i A B (5.4), we have which gives the ith row of AB as the product of the ith row of A with B. Proof Usig A i (A i ) to deote the ith row (colum) of A, we ca write eq(5.3) as the dot product c i A B i The ith row of C is therefore 2 p, 2,, Ai B, Ai B,, Ai B C c c c i i i ip Ai B 2 p Ai B, B,, B QED. Theorem 5.. Product Formula for Determiats. For ay two matrices A ad B, we have AB A B det det det (5.5) Proof By Lemma 5.0, we have AB d AB AB AB d A B A B A B det,,, 2 For a fixed B, we defie the fuctio det f A AB,,, 2

22 I terms of the rows of A, this becomes,,,,, f A A d A B A B A B 2 It is obvious that f satisfies Axioms ad 2 of a determiat fuctio. f I d B Furthermore, so that by the uiqueess Theorem 5.9, we have f A d A f I d Ad B QED.

23 5.2. The Determiat of the Iverse of a Nosigular Matrix We recall that a square matrix A is osigular if it has a left iverse B such that BA I. It is the show that B, if exists, is uique ad also a right iverse, i.e., AB I. Therefore, we call B the iverse of A ad deote it by Theorem 5.2. If matrix A is osigular, the det A 0 Proof det A ad (5.6) det A Usig the product formula (5.5), we have AA A A det det det det I QED. Commet Theorem 5.2 shows that det A 0 A. is a ecessary coditio for A to be osigular. Later, i Theorem 5.6, it will be show that it is also a sufficiet coditio, i.e., det A 0 implies the existece of A.

24 5.3. Determiats ad Idepedece of Vectors Theorem 5.3. A set of vectors A,, A i -space is idepedet iff d A,, 0 A. Proof d A,, 0 A, The egatio of Theorem 5.8, which says depedece implies gives d A A implies A A,, 0,, is idepedet, thus provig the if part of the theorem. To prove the oly if part, let V be the liear -space i questio. Sice A,, A is idepedet, it forms a basis for V. By Theorem 4.2, there exists a liear trasformatio T : V V such that T A I for,, where I is the th uit coordiate vector. I terms of compoets i the I basis, we have t A I i i Settig B bi t i, this becomes A B I for,, where A ad I are row matrices. By Lemma 5.0, we have A B AB is a matrix with rows A,, A. Hece,, where A AB I for,, which meas AB I, i.e., A is osigular ad by Theorem 5.2, det A 0. QED.

25 5.4. The Determiat of a Bloc-Diagoal Matrix Let A ad B be ad m m matrices, respectively. The m m matrix A O C O B (a) where each O deotes a matrix of zeros, is called a bloc-diagoal matrix with two diagoal blocs A ad B. [ Note that the two O s i eq(a) do ot have the same dimesios if m ]. A example is C with A 0 ad B Obviously, bloc diagoal matrices of arbitrary blocs ca be similarly costructed. Theorem 5.4. For ay square matrix A ad B, A O det det A det B O B (5.7) Proof Let A be ad B be m m. Let I be the uit matrix. The A O A O I O O B O I m O B so that by the product formula Theorem 5., we have A O A O I O det det det O B O I m O B A Now, det O O I m ca be regarded as a fuctio of the rows of A. It is the obvious that it satisfies Axioms ad 2 of a determiat fuctio. Furthermore, settig A I, we have det I O A O det I m O I. Hece, det det A m O I. m

26 I Similarly, det O O det B B. QED.

27 5.6. Expasio Formulas by Cofactors Every row A a a,, of a matrix A ca be expressed as a liear i i i combiatio of the uit vectors I,, I, i.e., A a I i i Sice determiats are liear i each row, we have d A d a I, i,, i, aid, I, (5.9) We ow itroduce A i as the matrix obtaied from A by replacig its ith row with the uit vector I. For example, for 3, we have 0 0 A a2 a22 a 23 a3 a32 a 33 A a2 a22 a 23 a3 a32 a A 3 a2 a22 a 23 a3 a32 a 33 Thus det A i d A,, I,, A i so that (5.9) ca be writte as det A a det A (5.23) i i where i ca be ay iteger betwee ad. The quatity det A i is call the cofactor of the elemet ai ad is deoted by cof a i det A (5.22) i so that (5.20) becomes Theorem 5.5: det A a cof a (5.24) i i Obviously, oe ca apply (5.23,4) to the cofactors repeatedly to reduce deta to a liear combiatio etirely of determiats of matrices E that are related to the uit matrix I by purely row permutatios. By Theorem 5.3, det E or for eve or odd permutatios, respectively. Thus, eqs(5.23,4) are expasio formulae for

28 evaluatig deta. More specifically, they are expasios by the ith row cofactors.

29 5.7. The Cofactor Matrix Defiitio: Cofactor Matrix. Give a matrix A a i, the matrix cof A cof ai i, is called the cofactor matrix of A. Theorem 5.6. For ay matrix A a i, we have t det A cof A A I (5.25) t where t is the traspose operatio so that A i a. Thus, if det A 0, the A i exists ad is give by t A cof A det A (5.26) Proof Usig Theorem 5.5, we expad deta by its th row cofactors as det A a cof a (5.27) Let B be a matrix equal to A except for the ith ( i ) row, which equals to the th row of A. Thus, Bi A B so that by Theorem 5.3(b), det B 0. Expadig detb by its ith row cofactors gives Usig det B bicof bi 0 (5.28) b i a ad cof bi cof ai for all eq(5.28) becomes acof ai 0 where i (5.29) which is ofte called the expasio by alie cofactors. Eqs(5.27,9) ca be combied to give

30 acof ai i det A (5.30) This proves (5.25) sice a cof a Theorem 5.7 i A cof A A square matrix A is osigular iff det A 0. Proof This is simply the corollary of Theorems 5.2 ad 5.6. t i

31 5.8. Cramer s Rule Theorem 5.8: Cramer s Rule. Cosider a system of liear equatios aix bi ( i,, with uows x,, x ). If the coefficiet matrix A a i is osigular, the system has a uique solutio give by x b cof a for,, (5.3) det A Proof The system ca be writte as a matrix equatio AX B where ad,, t X x x osigular, there is a uique solutio X t A B cof A Commet,, t B b b are colum matrices. Sice A is B (5.32) det A x cof A b det A b cof a QED. det A Note that b cof a is the determiat of a matrix C obtaied from A by replacig the th colum with B. [See Theorem 5.20 for the formula for expasio by colum.] Thus, (5.3) ca be put ito the perhaps better ow form det C x det A

32 5.9. Expasio Formulas by Miors Defiitio: Mior Give a square matrix A of order 2, the square matrix A of order obtaied by deletig the th row ad th colum of A is called the, mior of A. Example A matrix A a i of order 3 has 9 miors, oe for each elemet. The miors of the st row are A a a a a A 2 a a a 2 23 a 3 33 A 3 a a a 2 22 a 3 32 The expasio (5.2) ca therefore be writte as det A a det a a det A a det A Theorem 5.9: Expasio by th Row Miors. For a square matrix A of order 2, the cofactors ad miors are related by cof a det det A (5.33) A Therefore, the expasio of deta by the th row miors is give by det (5.34) det A a A Proof We shall demostrate the validity of (5.33) by a series of progressively more geeral cases. To begi, cosider the case where a2 a22 a23 a 2 A a3 a32 a33 a 3 a a2 a3 a It is obvious that A ca be trasformed ito

33 a22 a23 a 2 0 A 0 a32 a33 a 3 0 a2 a3 a O O A by addig a ( row ) to the th row for each 2,,. Sice these are elemetary row operatios of type (3) [see 5.5], we have det A det A. 0 Furthermore, sice 0 A is bloc diagoal, we have det A det A. Hece, (5.33) 0 is verified for this case. Next, similar cosideratio for the case ad gives 0 0 a2 a2 a 2 A a 3 a3 a 3 a a a 0 0 a2 0 a 2 0 A a3 0 a3 a 0 a with det A det A. Also 0 A a a a a a a a a a a a a We ow regard det A 0 as a fuctio of the rows of the mior A ad write. It is obvious that 0 det A f A f A satisfies Axioms ad 2 [see 5.3] for a determiat fuctio of order. f A f I det A By the uiqueess Theorem 5.6, we have where I is the uit matrix of order. Note that the diagoal elemets of A are a, for 2,, ad a for,,. Settig A J i

34 0 A gives f I det det O O I O O O O I I O O det O O det I O O I which verifies (5.35) for the case ad. Now, the geeral case of arbitrary ca be obtaied from the case by movig the th row to the st. This itroduces a extra factor so that (5.33) is verified. Fially, puttig (5.33) ito (5.24) gives (5.34). QED.

35 5.2. Existece of the Determiat Fuctio Apostle treated Theorem 5.20 as a existece theorem for the determiat fuctio. However, this existece was already demostrated by the expasio by row formulae (5.20) ad (5.34). Therefore, it seems more meaigful to treat Theorem 5.20 as a formula for a expasio by colum. Theorem Assume determiats of order exist. matrix A a i such that Let f be a fuctio of the rows of a f A,, A a det A (5.37) The f satisfies all 3 axioms for a determiat fuctio of order. Therefore, by iductio, determiats of arbitrary order exists. Proof By defiitio, the mior A is a matrix of order. Hece, det A satisfies all 3 axioms for a determiat fuctio of order. Now,,, det det t f A A f ta ta A a t A so that f satisfies axiom. row i to row. Thus, so that B Ai A ad B A for all b a a ad b a for all i,, Next, cosider the matrix B obtaied from A by addig Furthermore, sice det B are of order, they obey axiom 2. Deotig the rows with the st elemet trucated by a ~, we have det B f, A,, A A, det A for all i i i i o A det B f, Ai,, det A i o A

36 det Bi f, Ai A, f, A, f, A i, det A i o Ai det A i o Ai o A where the factor i comes from movig A i from the th to the ith row. Hece, f B,, B b det B i b det B b det B b det B i i i i i a a det A a det A det A i i i i a det A i a det A a det A a det A f A,, A i i i so that f satisfies axiom 2. Fially, settig A I gives f I,, I a det A I det so that axiom 3 is also satisfied. QED. Commet The foregoig proof is easily adapted to show that f A,, A a det A (5.38) for ay,,. Furthermore (5.38) is a colum expasio formula. This is to be cotrasted with the row expasio formula (5.34). Theorem 5.2. Proof det t A det A The proof is by iductio. Sippig the trivial case of, we see that the case 2 is easily verified usig eq(5.). Next, we assume the theorem is true for case

37 ad try to prove case. t To begi, let A a i ad B A b i, where bi a i. Expadig deta ad detb by the st colum ad row expasios, respectively, we have det det A a A det a det B det B b B Sice A ad B are order matrices, we have det B t det A det A det A det B QED.

38 5.23. Summary Defiitio: Determiat Fuctio The determiat fuctio of order is defied as R C by A A d A A d : M, or such that the followig axioms are satisfied: Axiom. Homogeeity I Each Row:,,,, i i det,, d ta td A where t R or C. Axiom 2. Axiom 3. Ivariace Uder Row Additio: Scale:, i,,,, i,,, d A A A d A A det,, I d I I Theorem 5.: Zero Vector If some row of A is the zero vector, the det A Theorem 5.2., i,,,, i,,, d A ta A d A A where i ad t is a scalar Theorem 5.3: Atisymmetry (a) d, Ai,, A, d, A,, Ai, for all i. (b) d, A,, A, 0 i i 5.4. Theorem 5.4: Diagoal Matrix If A diag a a a, 22,,, the det A a a a 22 a (5.4) i ii

39 5.5. Theorem 5.5: Upper Matrix For a upper diagoal matrix U, detu u u u Upper Matrix p u (5.7) i detu c c det A q det A c A det I (5.8) ii 5.7. Theorem 5.6. Uiqueess Theorem For Determiats f A d A f I (5.9) 5.9. Theorem 5.7. Additivity I Each Row.,,,,,, d A V d A d V (5.0a) Theorem 5.8: Depedece If the rows of a matrix A are depedet, the d A Theorem 5.9: Expasio Formula a a a 2 a 2 22 a a a a (5.2a) a a a 2 3 a a a a a a a a a a a a a a a (5.2) a32 a33 a3 a33 a3 a Lemma 5.0. AB i i A B (5.4)

40 5.. Theorem 5.. Product Formula for Determiats. AB A B det det det (5.5) 5.2. Theorem 5.2: Iverse det A (5.6) det A 5.3. Theorem 5.3: Idepedece A set of vectors A,, A i -space is idepedet iff 5.4. Theorem 5.4: Bloc Matrix d A,, 0 A. A O det det det O B A B (5.7) 5.5. Expasio by the ith row cofactors det A a cof a (5.24) cof a i i i det A i d A,,,, I A (5.22) 5.7. Defiitio: Cofactor Matrix. cof A cof ai i, 5.7. Theorem 5.6: Expasio by Cofactors t det A cof A If det A 0, the A I (5.25) t A cof A det A (5.26) 5.7. Theorem 5.7: Nosigularity A square matrix A is osigular iff det A 0.

41 5.8. Theorem 5.8: Cramer s Rule. If A a i is osigular, the system aix bi ( i,, has a uique solutio give by x det A ) b cof a for,, (5.3) 5.9. Defiitio: Mior The square matrix A of order obtaied by deletig the th row ad th colum of A is called the, mior of A Theorem 5.9: Expasio by th Row Miors. cof a det det A (5.33) A The expasio of deta by the th row miors is det (5.34) det A a A 5.2. Theorem 5.20: Expasio by st Colum Miors. Assume determiats of order exist. Let f A,, A a det A (5.37) The determiats of arbitrary order exists Theorem 5.2: Traspose det t A det A