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# METH APPL MATH I MATH 601

Texas A&M

GPA 3.6

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CHAPTER DETERMINANTS With each square matn x it is possible to associate a real number called the deter minant of the matrix The value of this number will tell us whether the matrix is singular In Section 1 the de nition of the determinant of a matrix is given In Section 2 we study properties of determinants and derive an elimination method for evaluating determinants The elimination method is generally the simplest method to use for evaluating the determinant of an n x n matrix when n gt 3 In Section 3 we see how determinants can be applied to solving n x 11 linear systems and how they can be used to calculate the inverse of a matrix An application involving cryptography is also presented in Section 3 Further applications of determinants are presented in Chapters 3 and 6 THE DETERMINANT OF A MATRIX With each n x n matrix A it is possible to associate a scalar detA whose value will tell us whether the matrix is nonsingular Before proceeding to the general de nition let us consider the following cases CASE I I x I Matrices If A a is a 1 x 1 matrix then A will have a multiplicative inverse if and only if a 9E 0 Thus if we de ne detA a V then A will be nonsingular if and only if detA 0 IOO Chapter 2 Determinants CASE 2 2 X 2 Matrices Let A 111 112 021 022 By Theorem 143 A will be nonsingular if and only if it is row equivalent to I Then if all 7e 0 we can test whether A is row equivalent to I by performing the following operations 1 Multiply the second row of A by an all 12 115121 111122 2 Subtract 12 times the rst row from the new second row an 112 0 111022 aziaiz Since all 7E 0 the resulting matrix will be row equivalent to I if and only if 1 111022 1211112 95 0 If a 0 we can switch the two rows of A The resulting matrix 121 6122 0 112 will be row equivalent to I 39if and only if man 96 0 This requirement is equiv alent to condition 1 when an 0 Thus if A is any 2 X 2 matrix and we de ne dWA a11a22 0121121 then A is nonsingular if and only if detA a5 0 Notation We can refer to the determinant of a speci c matrix by enclosing the array between vertical lines For example 39 111 l3 4 then 2 1 represents the determinant of A Section The Determinant of 3 Matrix 39039 CASE 3 3 x 3 Matrices We can test whether a 3 x 3 matrix is nonsingular by performing row operations to see if the matrix is row equivalent to the identity matrix I Toquot carry out the elimination in the rst column of an arbitrary 3 x 3 matrix A let us rst assume that an 75 0 The elimination can then be performed by subtracting 021011 times the rst row from the second and 1311111 times the rst row from the third an an 1113 a 1 a 12 13 0 anazzazlaiz a11a23 a21a13 021 G22 G23 gt an all 131 132 133 0 allay 311112 11033 a31013 an an The matrix on the right will be row equivalent to I if and only if a11122 a21a12 011423 021113 111 1111 an 95 0 011032 a31012 a11133 131013 11 11 Although the algebra is somewhat messy this condition can be simpli ed to 2 anazzaas anaazaza a12a21a33 11211311123 alsaziasz a13031a22 96 0 Thus if we de ne 3 detA a11a22a33 a11a321123 61120214133 a12a31a23 113021032 0136131422 then for the case an 96 0 the matrix will be nonsingular if and only if detA 5e 0 What if a1 0 Consider the following possibilities i an 01121 0 ii a11 1121 01131 0 iii 111 c121 H31 0 In case i it is not dif cult to show that A is row equivalent to I if and only if 112021a33 a12a316123 a13aziasz a13a31a22 7E 0 But this condition is the same as condition 2 with all 0 The details of case i are left as an exercise for the reader see Exercise 7 In case ii it follows that 0 6112 113 A 0 122 123 131 6132 ass 2 Chapter 2 Determinants is row equivalent to I if and only if f13101121123 41221113 95 0 Again this is a special case of condition 2 with an 2 0221 0 Clearly in case iii the matrix A cannot be row equivalent to I and hence must be singular In this case if we set all 121 and a3 equal to 0 in formula 3 the result will be detA 0 In general then formula 2 gives a necessary and suf cient condition for a 3 x 3 matrix A to be nonsingular regardless of the value of all We would now like to de ne the determinant of an n X n matrix To see how to do this note that the determinant of a 2 x 2 matrix a a A 11 12 121 1122 can be de ned in terms of the two 1 x 1 matrices M11 122 and M12 121 The matrix M11 is formed from A by deleting its rst row and rst column and M12 is formed from A by deleting its rst row and second column The determinant of A can be expressed in the form 4 d6tA 1111122 alzazl a11 d6tM 11 1112 detM12 For a 3 x 3 matrix A we can rewrite equation 3 in the form detA 611101221133 1321123 11201211133 11311123 013a21a32 131022 For j 12 3 let M1 1 denote the 2 x 2 matrix formed from A by deleting its rst row and jth column The determinant of A can then be represented in the form 5 detA all detM11 1112 detM 12 1113 detM13 where a 2 0123 121 023 121 122 M 11 2 M 12 M13 an 033 131 1133 a31 1132 To see how to generalize 4 and 5 to the case n gt 3 we introduce the following de nition 39 D DEFlNlTION Let A aij be an n x n matrix and let Mij denote the n l x n 1 matrix obtained from A by deleting the row and column containing ii The determinant of Mij is called the mirror of aij We de ne the cofactor Ag of a by AU lij detMj lt Section I The Determinant of a Matrix I03 In View of this de nition for a 2 x 2 matrix A we may rewrite equation 4 in the form 5 detA anArr 1121412 n 2 Equation 6 is called the cofactor expansion of detA along the rst row of A Note that we could also write 7 detA 12101112 a22a11 a211421 1221422 Equation 7 expresses detA in terms of the entries of the second row of A and their cofactors Actually there is no reason why we must expand along a row of the matrix the determinant could just as well be represented by the cofactor expansion along one of the columns d6tA a11a22 a21a12 a111 111 a21421 rst column detA aid 1121 a22a11 a12 112 1122142 second column For a 3 x 3 matrix A we have 8 detA a11A11a12A12l113A13 Thus the determinant of a 3 X 3 matrix can be de ned in terms of the elements in the rst row of the matrix and their corresponding cofactors EXAMPLE I If 254 A312 546 detM 11111411 1121412 113113 12a11d6tM11 13a12detMu lt 14a13 detM13 12 32 31 2 46 5 5 6 4 4395 4 26 8 S18 10412 5 16 4 As in the case of 2 x 2 matrices the determinant of a 3 x 3 matrix can be repre sented as a cofactor expansion using any row or column For example equation 3 can be rewritten in the form I04 Chapter 2 Determinants detA alzdaiazs alaaaiazz a11032 123 a13a21a32 a1111221133 aizaziasa a31a12a23 6131022 61320111023 M3021 41330111022 a12a21 a31A31 11321432 1331433 This is the cofactor expansion along the third row of A EXAMPLE 2 Let A be the matrix in Example 1 The cofactor expansion of detA along the second column is given by 32 24 24 detA 5 5615639 4l32 518 10112 20 44 12 16 4 The determinant of a 4 X 4 matrix can be de ned in terms of a cofactor expan sion along any row or column To compute the value of the 4 X 4 determinant we would have to evaluate four 3 x 3 determinants b DEFINITION The determinant of an n X n matrix A denoted detA is a scalar associated with the matrix A that is de ned inductively as follows an if n 1 detA a11A11a12A12 11Aln if n gt 1 where Auew mmm jhum are the cofactors associated with the entries in the rst row of A 4 As we have seen it is not necessary to limit ourselves to using the rst row for the cofactor expansion We state the following theorem without proof P THEOREM 2 I I If A is an n xn matrix with n z 2 then detA can be expressed as a cofactor expansion using any row or column of A detA anAii aiZAiZ 39 ainAin aleij aleZj quot 4 anjAnj fari1nandj1n Q The cofactor expansion of a 4 X 4 determinant will involve four 3 x 3 determi nants We can often save work by expanding along the row or column that contains the most zeros For example to evaluate O 2 3 0 0 4 5 0 0 1 0 3 2 0 l 3 Section The Determinant of a Matrix I05 we would expand down the rst column The rst three terms will drop out leaving 2 3 0 2 24 5 0 23 3l12 4 5 10 3 The cofactor expansion can be used to establish some important results about determinants These results are given in the following theorems e THEOREM 242 IfA is an n x n matrix then detAT detA Proof The proof is by induction on n Clearly the result holds if n 1 since a 1 X 1 matrix is necessarily symmetric Assume that the result holds for all k X k matrices and that A is a k 1 X k 1 matrix Expanding detA along the rst row of A we get detA a11detM11 012 detMiz i a1k1detM1k1 Since the Mij s are all k X k matrices it follows from the induction hypothesis that 9 detA an detMlTl an detMsz i gum detMf k 1 The right hand side of 9 is just the expansion by minors of detAT using the rst column of AT Therefore detAT detA 4 9 THEOREM 2 t 3 If A is an n X n triangular matrix the determinant of A equals the product of the diagonal elements of A Proof In view of Theorem 212 it suf ces to prove the theorem for lower trian gular matrices The result follows easily using the cofactor expansion and induction on n The details of this are left for the reader see Exercise 8 V THEOREM 2l4 Let A be an n X n matrix i If A has a row or column consisting entirely of zeros then detA 0 ii If A has two identical rows or two identical columns then detA O 4 Both of these results can be easily proved using the cofactor expansion The proofs are left for the reader see Exercises 9 and 10 In the next section we look at the effect of row operations on the value of the determinant This will allow us to make use of Theorem 213 to derive a more ef cient method for computing the value of a determinant 1 Given l06 Chapter 2 Determinants a Find the values of detM21 detMzz and detM23 b Find the values of A21 A22 and A23 0 Use your answers from part b to compute detA 2 Use determinants to determine whether the following 2 x 2 matrices are nonsin gular 3 5 3 6 3 6 all 1 04 l 2 4 2 4 2 4 3 Evaluate the following determinants l 3 S 5 2 a 2 3 b l8 4 c 2 4 5 4 5 4 3 O 1 3 2 2 1 2 d 3 l 2 e 4 1 2 f 1 3 2 5 1 4 2 1 3 5 1 6 2 0 39 O 1 2 1 2 1 0 1 O 0 3 0 1 1 g 1 6 2 o h 1 2 2 1 l l 2 3 3 2 3 l 4 Evaluate the following determinants by inspection 35 2 o o 23 a b 4 1 0 c 2 l 1 d 2 4 7 3 2 1 2 2 2 O 3 4 1 0 2 3 5 Evaluate the following determinant Write your answer as a polynomial in x a x b c 6 Find all values of A for which the following determinant will equal 0 1 2 t 4 3 3 A 7 Let A be a 3 x 3 matrix with all O and 121 3A 0 Show that A is row equivalent to I if and only if a121121033 arzasra23 a13a21a32 11311310122 79 0 8 Write out the details of the proof of Theorem 213 Section 2 Properties of Determinants I07 9 Prove that if a row or a column of an n x n matrix A consists entirely of zeros then detA O 1 Use mathematical induction to prove that if A is an n 1 X n 1 matrix with two identical rows then detA 0 11 Let A and B be 2 X 2 matrices a Does detA B detA detB b Does detAB detA detB c Does detAB detBA Justify your answers 12 Let A and B be 2 x 2 matrices and let C 111 112 b21 1722 D b b 11 12 l E 121 122 0 oz 3 0 a Show that detA B detA detB detC detD b Show that if B EA then detA B detA detB 13 Let A be a symmetric tridiagonal matrix ie A is symmetric and aij 0 whenever i j gt 1 Let B be the matrix formed from A by deleting the rst two rows and columns Show that detA an detMu 12 detB PROPERTIES OF DETERMINANTS In this section we consider the effects of row operations on the determinant of a matrix Once these effects have been established we will prove that a matrix A is singular if and only if its determinant is zero and we will develop a method for evaluating determinants using row operations Also we will establish an important theorem about the determinant of the product of two matrices We begin with the following lemma P LEMMA 22 Let A be an n X n matrix If Ajk denotes the cafactor of ajk for k1 n then detA i i 39 1 aiiAj1ai2Aj239ainAjn f J 0 ifi aej Proof If i j l is just the cofactor expansion of detA along the ith row of A To prove 1 in the case 1 j let A be the matrix obtained by replacing the jth row of A by the ith row of A 08 Chapter 2 Determinants all 412 39 39 39 In a a39 a 11 12 m Jth row A 39 ail ail in anl an2 39 39 arm Since two rows of A are the same its determinant must be zero It follows from the cofactor expansion of detA along the jth row that 0 detm HA 21172 39 ainAfn ai1Aj1ai2Aj2 ainAjn Q Let us now consider the effects of each of the three row operations on the value of the determinant Row Operation Two rows of A are interchanged IfAisa2x2matrixand O l E 1 0 then detEA a2 22 a21a12 and detA all 12 For n gt 2 let Eij be the elementary matrix that switches rows 139 and j of A It is a simple induction proof to show that detEjA detA We illustrate the idea behind the proof for the case n 3 Suppose that the rst and third rows of a 3 x 3 matrix A have been interchanged Expanding detE13A along the second row and making use of the result for 2 x 2 matrices we see that 1131 I32 133 detE13A 0121 1122 023 an an L113 32 a 1131 133 031 6131 021 1 quot a 12 013 an 6113 all 112 112 113 611 113 all a12 a2 02 123 132 133 131 1133 131 132 Section 2 Properties of Determinants I09 In general if A is an n x n matrix and Eij is the n x n elementary matrix formed by interchanging the ith and jth rows of I then detEiJA detA In particular detEij detEiJI detI 1 Thus for any elementary matrix E of type I detEA detA detE detA Row Operation ii A row of A is multiplied by a nonzero constant Let E denote the elementary matrix of type 11 formed from I by multiplying the ith row by the nonzero constant a If detEA is expanded by cofactors along the ith row then detEA mum1 aaizAiz vialquotAm ailAil aizAiz ainAln a detA In particular detE detEI udetI a and hence detEA 0 detA detE detA Row Operation lll A multiple of one row is added to another row Let E be the elementary matrix of type III formed from I by adding c times the ith row to the jth row Since E is triangular and its diagonal elements are all 1 it follows that detE 1 We will show that detEA detA detE detA If detEA is expanded by cofactors along the j th row it follows from Lemma 221 that detEA aj1cai1Aj1aj2cai2Ajz 39 ajn CainAjn aj1Aji ajnAjn CailAj1 ainAjn detA Thus detEA detA detE detA 0 Chapter 2 Determinants Summary In summation if E is an elementary matrix then detEA detE detA where 1 ifE is oftypel 2 detE a g 0 if E is of type II 1 if E is of type 111 Similar results hold for column operations Indeed if E is an elementary matrix then detAE detAET detETAT detET detAT detE detA Thus the effects that row or column operations have on the value of the determinant can be summarized as follows H Interchanging two rows or columns of a maln39x changes the sign of the determinant II Multiplying a single row or column of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar III Adding a multiple of one row or column to another does not change the value of the determinant NOTE As a consequence of H1 if one row or column of a matrix is a multiple of another the determinant of the matrix must equal zero Main Results We can now make use of the effects of row operations on determinants to prove two major theorems and to establish a simpler method of computing determinants It follows from 2 that all elementary matrices have nonzero determinants This observation can be used to prove the following theorem 9 THEOREM 222 An n x n matrix A is singular if and only if detA 0 gtProof The matrix A can be reduced to rowechelon form with a nite number of row operations Thus U EkEk139 39 ElA where U is in row echelon form and the Ei s are all elementary matrices detU detEkEk1 mm detEk detEk1 detEl detA Section 2 Properties of Determinants I I l Since the determinants of the Ei s are all nonzero it follows that detA 0 if and only if detU 0 If A is singular then U has a row consisting entirely of zeros and hence detU 0 If A is nonsingular U is tn39angular with 1 s along the diagonal and hence detU 1 From the proof of Theorem 222 we can obtain a method for computing detA Reduce A to row echelon form U EkEk1 E1A If the last row of U consists entirely of zeros A is singular and detA 0 Otherwise A is nonsingular and detA detEk detEk1 detE1 l Actually if A is nonsingular it is simpler to reduce A to triangular form This can be done using only row operations I and HI Thus T EmEm l 39 39 EIA and hence detA ldetT 3111222 run where the Iii s are the diagonal entries of T The sign will be positive if row operation I has been used an even number of times and negative otherwise EXAM PLE I Evaluate 2 1 3 4 2 1 6 3 SOLUTION 2 1 3 2 1 3 2 1 3 4 2 1 0 0 5 10 6 5 6 3 4 0 6 5 0 05 12 65 60 4 We now have two methods for evaluating the determinant of an n x n matrix A If n gt 3 and A has nonzero entries elimination is the most ef cient method in the sense that it involves fewer an39thmetic operations In Table l the number of arithmetic operations involved in each method is given for n 2 3 4 5 10 It is not dif cult to derive general formulas for the number of operations in each of the methods see Exercises 17 and 18 We have seen that for any elementary matrix E detEA detE detA detAE This is a special case of the following theorem HZ Chapter Determinants Cofactors Elimination Multiplications n Additions Multiplications Additions and Divisions 2 1 2 1 3 3 5 9 5 10 4 23 40 14 23 5 1 19 205 30 45 10 3628799 6235300 285 339 F THEOREM 223 If A and B are n X n matrices then detAB detA detB PPmnf If B is singular it follows from Theorem 143 that AB is also singular see Exercise 14 of Chapter 1 Section 4 and therefore detAB 0 detA detB If B is nonsingular B can be written as a product of elementary matrices We have already seen that the result holds for elementary matrices Thus detAB detAEkEk1 E1 detA detEk detEk1 detE1 detA detEkEk1 E1 detA detB 4 If A is singular the computed value of detA using exact arithmetic must be 0 However this result is unlikely if the computations are done by computer Since computers use a nite number system roundoff errors are usually unavoid able Consequently it is more likely that the computed value of detA will only be near 0 Because of roundoff errors it is virtually impossible to determine computa tionally whether a matrix is exactly singular In computer applications it is often more meaningful to ask whether a mam x is close to being singular In general the value of detA is not a good indicator of nearness to singularity In Section 5 of Chapter 6 we will discuss how to determine whether a matrix is close to being singular 1 Evaluate each of the following determinants by inspection 1113 0001 003 0311 1000 202 0022 0100 1 1 12 0010 Section 2 Properties of Determinants I l3 2 Let 0 l 2 3 l l 1 l A 2 2 3 3 1 2 2 3 a Use the elimination method to evaluate detA b Use the value of detA to evaluate O 1 2 3 0 1 2 3 2 2 3 3 1 1 l l 1 2 2 3 1 1 4 4 1 l l 1 2 3 1 2 3 For each of the following compute the determinant and state whether the matrix is singular or nonsingular 331 31 31 a b c012 62 42 023 1111 211 2 13 2132 d 4 3 5 e 1 2 2 f 0121 212 140 0 0 7 3 4 Find all possible choices of c that would make the following matrix singular 1 l l l 9 c 1 c 3 5 Let A be an n x n matrix and a a scalar Show that detaA 0quot detA 6 Let A be a nonsingular matrix Show that 1 det A 1 detA 7 Let A and B be 3 x 3 matrices with detA 4 and detB 5 Find the value of a detAB b det3A c det2AB d detA1B 4 pa Chapter 2 Determinants 8 Let E1 E2 E3 be 3 x 3 elementary matrices of types I H and III respectively and let A be a 3 X 3 matrix with detA 6 Assume additionally that E2 was formed from I by multiplying its second row by 3 Find the values of each of the following a detElA d detAEl b detEzA e detElz C detE3A f detElEzEa G Let A and B be row equivalent matrices and suppose that B can be obtained from A using only row operations I and 111 How do the values of detA and detB compare How will the values compare if B can be obtained from A using only row operation Ill Explain your answers Consider the 3 X 3 Vandermonde matrix 1 x1 x12 V 1 x2 x 1 x3 xg a Show that detV x2 x1x3 x1x3 x2 Hint Make use of row operation 111 b What conditions must the scalars x1 x2 x3 satisfy in order for V to be nonsingular 1 Suppose that a 3 x 3 matrix A factors into a product 1 O 0 M11 M12 u13 In 1 0 0 M22 M3 131 132 1 0 0 H33 Determine the value of detA 12 Let A and B be n x n matrices Prove that the product AB is nonsingular if and only if A and B are both nonsingular 13 Let A and B be n x n matrices Prove that if AB I then BA I What is the signi cance of this result in terms of the de nition of a nonsingular matrix 14 A matrix A is said to be skew symmetric if AT A For example 0 1 A 1 0 is skew symmetxic since Section 3 Cramer s Rule I I5 Show that if A is an n x n skew symmetric matrix and n is odd then A must be singular 15 Let A be a nonsingular n x n matrix with a nonzero cofactor Am and set detA C Arm Show that if we subtract 6 from Am then the resulting matrix will be singular 16 Let x and y be elements of R3 and let 2 be the vector in R3 whose coordinates are de ned by x2 X3 x1 x3 X1 x2 11 a 12 Y7 3 3 yl Ya YI J 2 Let X x x yT and Y x y yT Show that xTz detX 0 and yTz detY 0 17 Show that evaluating the determinant of an n x n matrix by cofactors involves n 1 n 1 additions and vak multiplications 1 a Show that the elimination method of computing the value of the determinant of an n xn matrix involves nn l2n 16 additions and n 1n2n33 multiplications and divisions Hint At the ith step of the reduction process it takes n i divisions to calculate the multiples of the ith row that are to be subtracted from the remaining rows below the pivot We must then calculate new values for the n i2 entries in rows i 1 thr0ugh n and columns i 1 through n CRAMER S RULE In this section we learn a method for computing the inverse of a nonsingular matrix A using determinants We also learn a method for solving Ax b using determinants Both methods depend on Lemma 221 The Adjoint of 1 Matrix Let A be an n x n matrix We de ne a new matrix called the adjoint of A by A11 A21 An A A A ade 12 22 n2 Aln A27 quot Ann Thus to form the adjoint we must replace each term by its cofactor and then trans pose the resulting matrix By Lemma 221 6 Chapter 2 Determinants detA if i 39 ailAjl ai2Aj2ainAjn J o if i as j and hence it fOIIOWS that Aadj A detAI If A is nonsingulax detA is a nonzero scalax and we may write 1 A detm ad A 1 1 1 detA adj A EXAMPLE 5 For a 2 X 2 matrix adj A a lm 021 1111 If A is nonsingular then A 1 an a12 auazz alzazi a21 an EXAMPLE 2 Let 212 A 322 123 Compute adj A and A l SOLUTEQN 2 32 32 1312 1 dA 22 217 31 1312 4 3 Section 3 Cramer s Rule I I7 2 2 1 1 d39 A rd detAaJ 5 7 4 2 4 3 1 Using the formula 1 1 dam ad A we can derive a rule for representing the solution to the system Ax b in terms of determinants P THEOREM 23 Cramer39s Rule Let A be an n X n nonsingular matrix and let b e Rquot Let Ai be the matrix obtained by replacing the ith column of A by b If X is the unique solution to Ax b then let A xidctA39 for il2n VProof Since 1 1 b d A b x detAaJ it follows that x blAli b2A2i quot bnAni l detA detA lt detA EXAMPLE 3 Use Cramer s rule to solve x1 2x2 X3 5 3 2x1 2x2 X3 6 x12xg3x3 9 SOLUTION l 2 l 5 2 1 detA 2 l 4 detA1 6 2 1 4 l 2 3 9 2 3 1 5 1 l 2 5 detA2 2 6 l 4 detA3 2 2 6 8 1 9 3 1 2 9 Therefore 4 4 x1 1 x2 1 X3 82 Q 8 Chapter 2 Determinants Cramer s rule gives us a convenient method for writing the solution to an n X n system of linear equations in terms of determinants To compute the solution however we must evaluate n 1 determinants of order n Evaluating even two of these determinants generally involves more computation than solving the system using Gaussian elimination Section 3 Cramer s Rule I I9 1 For each of the following compute i detA ii adj A and 111 A 1 2 31 aA bA I 3 1 2 4 1 3 1 111 cA 2 1 1 dA 0 2 2 1 1 2 Use Cramer s rule to solve each of the following systems a x12x23 b 2x13x22 C 2x1 X2 3X30 3x1 x21 3x12x25 4x15x2 3638 2x1 X24X32 d X13xzx3 1 e x1x2 0 2x1 x2x3 5 x2 X3 2JC41 2x12x2 x3 8 x1 2x3 X40 x1x2 1640 20 Chapter 2 Determinants 3 Given 121 A043 122 Determine the 2 3 entry of A 1 by computing a quotient of two determinants 4 Let A be the matrix in Exercise 3 Compute the third column of A by using Cramer s rule to solve Ax e3 5 Given 1 2 3 A 2 3 4 3 4 5 a Compute the determinant of A Is A nonsingular b Compute adj A and the product A adj A 6 If A is singular what can you say about the product A adj A 7 Let Bj denote the matrix obtained by replacing the jth column of the identity matrix with a vector b 111 bnT Use Cramer s rule to show that b detB 8 Let A be a nonsingular n x n matrix with n gt 1 Show that detadj A detA 1 9LetAbea4x4mattixIf 39 for jln 2 0 0 0 O 2 1 0 adj A 0 4 3 2 0 2 1 2 a Calculate the value of detadj A What should the value of dctA be Hint Use the result from Exercise 8 b Find A 10 Show that if A is nonsingular then adj A is nonsingular and adj Arl detA 1A adj A 1 11 Show that if A is singular then adj A is also singular 12 Show that if detA 1 then adj adj A A MATLAB Exercises 2 13 Suppose that Q is a matrix with the property Q 1 QT Show that Qij detQ 14 In coding a message a blank space was represented by 0 an A by l a B by 2 a C by 3 and so on The message was transformed using the matrix Qij l l 2 O 1 1 1 0 A O 0 l l l 0 O l and sent as 19 19 25 21 018 18 15 310 8 3 2 20 7 12 What was the message The rst four exercises involve integer matrices and illustrate some of the properties of determinants that were covered in this chapter The last two exercises illustrate some of the differences that may arise when we work with determinants in oating point arithmetic In theory the value of the determinant should tell us whether the matrix is nonsingular However if the matrix is singular and its determinant is computed using nite precision arithmetic then because of roundoff errors the computed value of the determinant may not equal zero A computed value near zero does not necessarily mean that the matrix is singular or even close to being singular Furthermore a matrix may be singular or nearly singular and have a determinant that is not even close to zero see Exercise 6 1 Generate random 5 x 5 matrices with integer entries by setting A round10 rand5 and B round20 rand5 10 Use MATLAB to compute each of the following pairs of numbers In each case check whether the rst is equal to the second a detA detAT b detA B detA detB c detAB detA detB d detAT 3T detAT detBT e detA 1 l detA f detA 13 1 detA detB 2 Are n x n magic squares nonsingular Use MATLAB to compute detmagicn in the cases 11 3 4 10 Whatseems to be happening Check the cases n 24 and 25 to see if the pattern still holds 5 Set A round10 rand6 In each of the following use MATLAB to compute a second matrix as indicated State how the second matrix is related I22 Chapter 2 Determinants to A and compute the determinants of both matrices How are the determinants related a B A B2 Al B1 A2 b CA C3 4A3 c D A D5 A5 2 gt1 A4 2 We can generate a random 6 x 6 matrix A whose entries consist entirely of zeros and ones by setting A roundrand6 a What percentage of these random 0 1 matrices are singular You can esti4 mate the percentage using MATLAB by setting y zeros1 100 and then generating 100 test matrices and setting y j 1 if the jth matrix is singular and 0 otherwise The easy way to do this in MATLAB is to use a for loop Generate the loop as follows for j 1 100 A roundrand6 yj detA 0 end Note A semicolon at the end of a line suppresses printout It is recom mended that you include one at the end of each line of calculation that occurs inside a for loop To determine how many singular matrices were generated use the MATLAB command sumy What percentage of the matrices generated were singular b For any positive integer n we can generate a random 6 x 6 matrix A whose entries are integers from 0 to n by setting A roundn I rand6 What percentage of random integer matrices generated in this manner will be singular if n 3 If n 6 If n 10 We can estimate the answers to these questions using MATLAB In each case generate 100 test matrices and determine how many of the matrices are singular If a matrix is sensitive to roundoff errors the computed value of its determinant may differ drastically from the exact value For an example of this set U round100 gtrlt rand10 U tr1uU 1 01 gt1 eye10 In theory detU detUT 10 10 and detUUT detU detUT 10 20 Chapter Test I 23 Compute detU detU and detU U using MATLAB Do the computed values match the theoretical values 6 Use MATLAB to construct a matrix A by setting A vanderl 6 A A diagsumA a By construction the entries in each row of A should all add up to zero To check this set x ones6 1 and use MATLAB to compute the product Ax The matrix A should be singular Why Explain Use the MATLAB functions det and inv to compute the values of detA and A39l Which MATLAB function is a more reliable indicator of singularity b Use MATLAB to compute detAT Are the computed values of detA and detAT equal Another way to check if a matrix is singular is to compute its reduced row echelon form Use MATLAB to compute the reduced row echelon forms of A and AT c To see what is going wrong it helps to know how MATLAB computes deter minants The MATLAB routine for determinants rst computes a form of the LU factorization of the matrix The determinant of the L is 1 depending on the whether an even or odd number of row interchanges were used in the computation The computed value of the determinant of A is the product of the diagonal entries of U multiplied by detL tl In the special case that the original matrix has integer entries the exact determinant should take on an integer value So in this case MATLAB will round its decimal answer to the nearest integer To see what is happening for our original matrix use the following commands to compute and display the factor U format short e LU 1uA U In exact arithmetic U should be singular Is the computed matrix U singular If not what goes wrong Use the following commands to see the rest of the computation of d detA format short 039 proddiagU d roundd In each of the following answer true if the statement is always true and false other wise In the case of a true statement explain or prove your answer In the case of a false statement give an example to show that the statement is not always true In each of the following assume that all the matrices are n x n 1 detAB detBA 2 detA B detA detB 3 detcA cdetA 4 detABT detA detB l24 Chapter 2 Determinants Squot detA detB implies A B detAk detAk OK 7 A triangular matrix is nonsiugular if and only if its diagonal entries are all nonzero 8 If x is a nonzero vector in Rquot and Ax 0 then detA 0 V If A and B are row equivalent matrices then their determinants are equal If A 75 0 but Ak 0 where 0 denotes the zero matrix for some positive integer k then A must be singular

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