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1.5 Elementary Matrices 1 2(g) 3 368 3 1 1 1(h) 3 101 2 1 1 311. Given 3A 5 1 2and 1B 3 2 4compute A 1 and use it to(a) find a 2 2 matrix X such that AX B.(b) find a 2 2 matrix Y such that Y A B. 3 , B 622 2 ,C 44 6where a0 , a1 , . . . , ak are scalars, thenAB B A. 2 3Solve each of the following matrix equations:(b) XA B C(a) AX B C(d) XA C X(c) AX B X13. Is the transpose of an elementary matrix an elementary matrix of the same type? Is the product of twoelementary matrices an elementary matrix?14. Let U and R be n n upper triangular matrices andset T UR. Show that T is also upper triangularand that t j j u j j r j j for j 1, . . . , n.15. Let A be a 3 3 matrix and suppose that2a1 a2 4a3 0How many solutions will the system Ax 0 have?Explain. Is A nonsingular? Explain.16. Let A be a 3 3 matrix and suppose thata1 3a2 2a3Will the system Ax 0 have a nontrivial solution?Is A nonsingular? Explain your answers.17. Let A and B be n n matrices and let C A B.Show that if Ax0 Bx0 and x0 0, then C mustbe singular.22. Show that if A is a symmetric nonsingular matrix,then A 1 is also symmetric.23. Prove that if A is row equivalent to B, then B isrow equivalent to A.24. (a) Prove that if A is row equivalent to B and B isrow equivalent to C, then A is row equivalentto C.(b) Prove that any two nonsingular n n matricesare row equivalent.25. Let A and B be m n matrices. Prove that if B isrow equivalent to A and U is any row echelon formA, then B is row equivalent to U .26. Prove that B is row equivalent to A if and onlyif there exists a nonsingular matrix M such thatB MA.27. Is it possible for a singular matrix B to be rowequivalent to a nonsingular matrix A? Explain.28. Given a vector x Rn 1 , the (n 1) (n 1)matrix V defined by 1if j 1vi j j 1for j 2, . . . , n 1xiis called the Vandermonde matrix.(a) Show that ifVc y18. Let A and B be n n matrices and let C AB.Prove that if B is singular, then C must be singular.[Hint: Use Theorem 1.5.2.]and19. Let U be an n n upper triangular matrix withnonzero diagonal entries.(a) Explain why U must be nonsingular.(b) Explain why U 1 must be upper triangular.then20. Let A be a nonsingular n n matrix and let B be ann r matrix. Show that the reduced row echelonform of (A B) is (I C), where C A 1 B.Leon:21. In general, matrix multiplication is not commutative (i.e., AB B A). However, in certain specialcases the commutative property does hold. Showthat(a) if D1 and D2 are n n diagonal matrices, thenD1 D 2 D 2 D 1 .(b) if A is an n n matrix andB a0 I a1 A a2 A 2 · · · ak A k12. Let 5A 367p(x) c1 c2 x · · · cn 1 x np(xi ) yi ,i 1, 2, . . . , n 1(b) Suppose that x1 , x2 , . . . , xn 1 are all distinct.Show that if c is a solution to V x 0, then thecoefficients c1 , c2 , . . . , cn must all be zero andhence V must be nonsingular.Linear Algebra with Applications 8/E 7/15/07 19:13 Page 67of

82thenChapter 1 Matrices and Systems of Equations 4AB 7 5 10and 11BA 4 7 3This proves thatreducedstatement (ii) is false.1. If the row echelon form of A involves free variables, then the system Ax b will have infinitelymany solutions.2. Every homogeneous linear system is consistent.3. An n n matrix A is nonsingular if and only if thereduced row echelon form of A is I (the identitymatrix).4. If A is nonsingular, then A can be factored into aproduct of elementary matrices.5. If A and B are nonsingular n n matrices, thenA B is also nonsingular and(A B) 1 A 1 B 1 .6. If A A 1 , then A must be equal to either I or I .7. If A and B are n n matrices, then(A B)2 A2 2AB B 2 .8. If AB AC and A O (the zero matrix), thenB C.9. If AB O, then B A O.10. If A is a 3 3 matrix and a1 2a2 a3 0, thenA must be singular.11. If A is a 4 3 matrix and b a1 a3 , then thesystem Ax b must be consistent.12. Let A be a 4 3 matrix with a2 a3 . If b a1 a2 a3 , then the system Ax b will haveinfinitely many solutions.13. If E is an elementary matrix, then E T is also anelementary matrix.14. The product of two elementary matrices is an elementary matrix.15. If x and y are nonzero vectors in Rn and A xyT ,then the row echelon form of A will have exactlyone nonzero row.CHAPTER TEST B1. Find all solutions of the linear systemx1 x2 3x3 2x4 1 x1 x2 2x3 x4 22x1 2x2 7x3 7x4 12. (a) A linear equation in two unknowns corresponds to a line in the plane. Give a similargeometric interpretation of a linear equation inthree unknowns.(b) Given a linear system consisting of two equations in three unknowns, what is the possiblenumber of solutions? Give a geometric explanation of your answer.(c) Given a homogeneous linear system consistingof two equations in three unknowns, how manysolutions will it have? Explain.3. Let Ax b be a system of n linear equations inn unknowns, and suppose that x1 and x2 are bothsolutions and x1 x2 .(a) How many solutions will the system have? Explain.(b) Is the matrix A nonsingular? Explain.4. Let A be a matrix of the form β αA 2α 2βLeon:where α and β are fixed scalars not both equal to 0.(a) Explain why the system 3 Ax 1must be inconsistent.(b) How can one choose a nonzero vector b so thatthe system Ax b will be consistent? Explain.5. Let 2 1 3 2 1 3 1 3 5 4 2 7 A B 4 2 71 3 5 0 1 3 0 2 7 C 5 3 5(a) Find an elementary matrix E such thatE A B.(b) Find an elementary matrix F such thatAF C.6. Let A be a 3 3 matrix and letb 3a1 a2 4a3Will the system Ax b be consistent? Explain.Linear Algebra with Applications 8/E 6/22/07 10:42 Page 82

152Chapter 3 Vector Spacesthe transition matrix from [1, 2x, 4x 2 2] to [1, x, x 2 ]. Since1 1 · 1 0x 0x 22x 0 · 1 2x 0x 24x 2 2 2 · 1 0x 4x 2 1 0S 0the transition matrix is 0 2 20 04The inverse of S will be the transition matrix from [1, x, x 2 ] to [1, 2x, 4x 2 2]: 1 0 12 1 S 1 00 2 1 0 0 4Given any p(x) a bx cx 2 in P3 , to find the coordinates of p(x) with respect to[1, 2x, 4x 2 2], we simply multiply 1 10c 2 a 11 b 0 2 0 b 2 c 1 1 c0 0 44Thus,p(x) (a 12 c) · 1 ( 12 b) · 2x 14 c · (4x 2 2)We have seen that each transition matrix is nonsingular. Actually, any nonsingularmatrix can be thought of as a transition matrix. If S is an n n nonsingular matrix and{v1 , . . . , vn } is an ordered basis for V , then define {w1 , w2 , . . . , wn } by (4). To see thatthe w j ’s are linearly independent, suppose thatn xjwj 0j 1It follows from (4) that nn i 1 si j x j v j 0j 1By the linear independence of the vi ’s, it follows thatn si j x j 0i 1, . . . , nj 1or, equivalently,Leon:Sx 0Linear Algebra with Applications 8/E 6/26/07 07:54 Page 43

248Chapter 5 OrthogonalityTheorem 5.5.7Let S be a subspace of an inner product space V and let x V . Let {u1 , u2 , . . . , un }be an orthonormal basis for S. Ifnp ci ui(3)i 1whereci x, ui for each i(4)then p x S (see Figure 5.5.2).xp–xSpFigure 5.5.2.ProofWe will show first that (p x) ui for each i: ui , p x ui , p ui , x n xi ,c j u j cij 1n c j ui , u j cij 1 0So p x is orthogonal to all the ui ’s. If y S, thennαi uiy i 1and hencennαi ui p x, y p x,i 1αi p x, ui 0i 1If x S, the preceding result is trivial, since, by Theorem 5.5.2, p x 0. Ifx S, then p is the element in S closest to x.Theorem 5.5.8Under the hypothesis of Theorem 5.5.7, p is the element of S that is closest to x; thatis, y x p x for any y p in S.Leon:Linear Algebra with Applications 8/E 7/15/07 14:34 Page 51

6.4 Hermitian Matrices2. Let 1 i 2z1 1 i 2 and i 2 z2 1 2(a) Show that {z1 , z2 } is an orthonormal set in C2 . as a linear 2 4i (b) Write the vector z 2icombination of z1 and z2 .3. Let {u1 , u2 } be an orthonormal basis for C2 , and letz (4 2i)u1 (6 5i)u2 .(a) What are the values of u1H z, z H u1 , u2H z, andz H u2 ?(b) Determine the value of z .4. Which of the matrices that follow are Hermitian?Normal? 2 i 1 1 i 2 (b) (a) 2 i 123 11 2 2 (c) 11 22 11 i 22 (d) 11 i 22 i1 0 0 2 i (e) i 1 2 i0 31 i i 1 i13 (f) i315. Find an orthogonal or unitary diagonalizing matrixfor each of the following: 13 i 2 1 (b)(a)3 i41 2 11 2 2 i 0 13 2 i 2 0 (d) (c) 1 230 0 2 0 0 1 1 1 1 1 1 1 0 1 0 (f) (e) 1 1 11 0 0 2 2 4 21 1 (g) 2 11Leon:3356. Show that the diagonal entries of a Hermitian matrix must be real.7. Let A be a Hermitian matrix and let x be a vectorin Cn . Show that if c xAx H , then c is real.8. Let A be a Hermitian matrix and let B i A. Showthat B is skew Hermitian.9. Let A and C be matrices in Cm n and let B Cn r .Prove each of the following rules:(a) (A H ) H A(b) (α A βC) H α A H βC H(c) (AB) H B HA H10. Let A and B be Hermitian matrices. Answer trueor false for each of the statements that follow. Ineach case, explain or prove your answer.(a) The eigenvalues of AB are all real.(b) The eigenvalues of AB A are all real.11. Show thatz, w w H zdefines an inner product on Cn .12. Let x, y, and z be vectors in Cn and let α and β becomplex scalars. Show thatz, αx βy α z, x β z, y 13. Let {u1 , . . . , un } be an orthonormal basis for a complex inner product space V , and letz a1 u1 a2 u2 · · · an unw b1 u1 b2 u2 · · · bn unShow thatnz, w bi aii 114. Given that15.16.17.18. 4 0A 001 i 0 i 1find a matrix B such that B H B A.Let U be a unitary matrix. Prove that(a) U is normal.(b) U x x for all x Cn .(c) if λ is an eigenvalue of U , then λ 1.Let u be a unit vector in Cn and define U I 2uu H . Show that U is both unitary and Hermitian and, consequently, is its own inverse.Show that if a matrix U is both unitary and Hermitian, then any eigenvalue of U must equal either 1or 1.Let A be a 2 2 matrix with Schur decompositionU T U H and suppose that t12 0. Show thatLinear Algebra with Applications 8/E 7/15/07 15:02 Page 54

318Chapter 6 Eigenvaluescolor-blind men is p and, over a number of generations, no outsiders have entered thepopulation, there is justification for assuming that the proportion of genes for colorblindness in the female population is also p. Since color blindness is recessive, wewould expect the proportion of color-blind women to be about p 2 . Thus, if 1 percentof the male population is color blind, we would expect about 0.01 percent of the femalepopulation to be color blind.The Exponential of a MatrixGiven a scalar a, the exponential ea can be expressed in terms of a power seriesea 1 a 1 21a a3 · · ·2!3!Similarly, for any n n matrix A, we can define the matrix exponential e A in terms ofthe convergent power series1 21A A3 · · ·(6)2!3!The matrix exponential (6) occurs in a wide variety of applications. In the case of adiagonal matrix λ1 λ2 D . . . λnthe matrix exponential is easy to compute: 1 m1 2De lim I D D · · · Dm 2!m! m 1 λ k λ1 e 1 k! λ2k 1 e . . lim . . . m . m 1 λn k e λ k! n k 1eA I A It is more difficult to compute the matrix exponential for a general n n matrix A. If,however, A is diagonalizable, thenfor k 1, 2, . . .Ak XD k X 1 11e A X I D D 2 D 3 · · · X 12!3! X e D X 1EXAMPLE 6 Compute e A forLeon: 2 6 A 13Linear Algebra with Applications 8/E 7/15/07 15:02 Page 37

6.5 The Singular Value DecompositionthenProof2 A A F σk 1 · · · σn2 1/2343 min A S FS MLet X be a matrix in M satisfying (7). Since A M, it follows that2 · · · σn2 A X F A A F σk 1We will show that2 A X F σk 1 · · · σn2 1/2(8) 1/2and hence that equality holds in (8). Let Q P T be the singular value decompositionof X , where ω1 ω2 O k . . .O O O ωk OOIf we set B Q T A P, then A Q P T , and it follows that A X F Q(B )P T F B Let us partition B in the same manner as B : k kk (n k)B11B12B21 B22 (m k) k F (m k) (n k)It follows that A X 2F B11 2k F B12 2F B21 2F B22 2FWe claim that B12 O. If not, then define B11Y Q O B12 PTOThe matrix Y is in M and A Y 2F B21 2F B22 2F A X 2FBut this contradicts the definition of X . Therefore, B12 O. In a similar manner, itcan be shown that B21 must equal O. If we set B11 O PTZ QO OLeon:Linear Algebra with Applications 8/E 7/15/07 15:02 Page 62

6.6 Quadratic Formsyyyx(i) Circley(iii) Hyperbola(ii) Ellipsexxx353(iv) ParabolaFigure 6.6.1.Case 3. The conic section has been rotated from the standard position by an angle θthat is not a multiple of 90. This occurs when the coefficient of the x y term is nonzero(i.e., b 0).In general, we may have any one or any combination of these three cases. To grapha conic section that is not in standard position, we usually find a new set of axes x andy such that the conic section is in standard position with respect to the new axes. Thisis not difficult if the conic has only been translated horizontally or vertically, in whichcase the new axes can be found by completing the squares. The following exampleillustrates how this is done:EXAMPLE 1 Sketch the graph of the equation9x 2 18x 4y 2 16y 11 0SolutionTo see how to choose our new axis system, we complete the squares:9(x 2 2x 1) 4(y 2 4y 4) 11 9 16This equation can be simplified to the form(y 2)2(x 1)2 12232If we letx x 1andthe equation becomesy y 2(x )2(y )2 12232which is in standard form with respect to the variables x and y . Thus, the graph, asshown in Figure 6.6.2, will be an ellipse that is in standard position in the x y -axissystem. The center of the ellipse will be at the origin of the x y -plane [i.e., at the point(x, y) (1, 2)]. The equation of the x -axis is simply y 0, which is the equationof the line y 2 in the x y-plane. Similarly, the y -axis coincides with the line x 1.Leon:Linear Algebra with Applications 8/E 7/15/07 15:02 Page 72

7.4 Matrix Norms and Condition Numbers11. Let A wyT , where w Rm and y Rn . ShowthatAx 2 y 2 w 2 for all x 0 in Rn .(a)x 2(b)12. LetA2 y2w2 3 1A 4(a) Determine A .(b) Find a vector x whose coordinates are each 1 such that Ax A . (Note thatx 1, so A Ax / x .)13. Theorem 7.4.2 states that nA max1 i m ai j j 1Prove this in two steps.(a) Show first thatnA max1 i m ai j j 1(b) Construct a vector x whose coordinates areeach 1 such that nAx Ax max ai j 1 i mx j 114. Show that A F A T F .15. Let A be a symmetric n n matrix. Show thatA A 1.16. Let A be a 5 4 matrix with singular values σ1 5, σ2 3, and σ3 σ4 1. Determine the valuesof A 2 and A F .17. Let A be an m n matrix.(a) Show that A 2 A F .(b) Under what circumstances will A 2 A F ?18. Let · denote the family of vector norms and let· M be a subordinate matrix norm. Show thatAM max Axx 119. Let A be an m n matrix and let · v and · wbe vector norms on Rn and Rm , respectively. ShowthatAx wA v,w maxx 0x vm ndefines a matrix norm on R .Leon:20. Let A be an m n matrix. The 1,2-norm of A isgiven byAx 2A 1,2 maxx 0x 1(See Exercise 19.) Show thatA 2 7 4 1214151,2 max ( a1 2 , a2 2 , . . . , an 2 )21. Let A be an m n matrix. Show thatA 1,2 A 222. Let A be an m n matrix and let B Rn r . Showthat(a) Ax A 1,2 x 1 for all x in Rn .(b) AB 1,2 A 2 B 1,223. Let A be an n n matrix and let · M be a matrix norm that is compatible with some vector normon Rn . Show that if λ is an eigenvalue of A, then λ A M .24. Use the result from Exercise 23 to show that if λ isan eigenvalue of a stochastic matrix, then λ 1.25. Sudoku is a popular puzzle involving matrices. Inthis puzzle, one is given some of the entries of a9 9 matrix A and asked to fill in the missing entries. The matrix A has block structure A11 A12 A13 A21 A22 A23 A A31 A32 A33where each submatrix Ai j is 3 3. The rules of thepuzzle are that each row, each column, and each ofthe submatrices of A must be made up of all of theintegers 1 through 9. We will refer to such a matrixas a sudoku matrix. Show that if A is a sudokumatrix, then λ 45 is its dominant eigenvalue.26. Let Ai j be a submatrix of a sudoku matrix A (seeExercise 25). Show that if λ is an eigenvalue of Ai j ,then λ 22.27. Let A be an n n matrix and x Rn . Prove:(a) Ax n 1/2 A 2 x (b) Ax 2 n 1/2 A x 2(c) n 1/2 A 2 A n 1/2 A 228. Let A be a symmetric n n matrix with eigenvalues λ1 , . . . , λn and orthonormal eigenvectorsu1 , . . . , un . Let x Rn and let ci uiT x fori 1, 2, . . . , n. Show thatn(a)Ax22 (λi ci )2i 1(b) If x 0, thenLinear Algebra with Applications 8/E 7/15/07 16:25 Page 30min λi 1 i nAx 2 max λi 1 i nx 2

474Answers to Selected Exercises5. (a) det(A) 0, so A is singular. 12 1 2 42 and(b) adj A 12 1 0 0 0 0 0 0 A adj A 0 0 09. (a) det(adj(A)) 8 and det(A) 2 1000 4 11 0 (b) A 0 62 2 010114. D O YOUR H OMEWORK .3.3CHAPTER TEST A1. True 2. False 3. False 4. True 5. False6. True 7. True 8. True 9. False 10. TrueChapter 33.1 1. (a) x1 10, x2 17(b) x3 13 x1 x2 2. (a) x1 5, x2 3 5 (b) x3 4 5 x1 x2 7. If x y x for all x in the vector space, then0 0 y y.8. If x y x z, then3.4 x (x y) x (x z)11.3.21.2.3.4.5.6.11.12.16.Leon:and the conclusion follows from axioms 1, 2,3, and 4.V is not a vector space. Axiom 6 does nothold.(a) and (c) are subspaces; (b), (d), and (e) arenot.(b) and (c) are subspaces; (a) and (d) are not.(a), (c), (e), and (f) are subspaces; (b), (d),and (g) are not.(a) {(0, 0)T }(b) Span(( 2, 1

Linear Algebra with Applications, 8th Ed. Steven J. Leon The following pages include all the items of errata that have been uncovered so far. In each case we include the entire page containing the errata and indicate the correction to be made. Help in uncover-ing additional errata would be gr