Exercises and Problems in Linear Algebra

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Exercises and Problems in Linear Algebra John M. Erdman Portland State University Version July 13, 2014 c 2010 John M. Erdman E-mail address: erdmanpdx.eduCHAPTER 1 SYSTEMS OF LINEAR EQUATIONS 1.1. Background Topics: systems of linear equations; Gaussian elimination (Gauss' method), elementary row op- erations, leading variables, free variables, echelon form, matrix, augmented matrix, Gauss-Jordan reduction, reduced echelon form. 1.1.1. De nition. We will say that an operation (sometimes called scaling) which multiplies a row of a matrix (or an equation) by a nonzero constant is a row operation of type I. An operation (sometimes called swapping) that interchanges two rows of a matrix (or two equations) is a row operation of type II. And an operation (sometimes called pivoting) that adds a multiple of one row of a matrix to another row (or adds a multiple of one equation to another) is arowoperation of type III. 34 1. SYSTEMS OF LINEAR EQUATIONS 1.2. Exercises (1) Suppose that L and L are lines in the plane, that the x-intercepts of L and L are 5 1 2 1 2 and1, respectively, and that the respective y-intercepts are 5 and 1. Then L and L 1 2 intersect at the point ( , ) . (2) Consider the following system of equations. 8 w +x +y +z = 6 w +y +z = 4 () : w +y = 2 (a) List the leading variables . (b) List the free variables . (c) The general solution of () (expressed in terms of the free variables) is ( , , , ) . (d) Suppose that a fourth equation2w +y = 5 is included in the system (). What is the solution of the resulting system? Answer: ( , , , ). (e) Suppose that instead of the equation in part (d), the equation2w 2y =3 is included in the system (). Then what can you say about the solution(s) of the resulting system? Answer: . (3) Consider the following system of equations: 8 x + y + z = 2 x + 3y + 3z = 0 () : x+ 3y+ 6z = 3 (a) Use Gaussian elimination to put the augmented coecient matrix into row echelon 2 3 1 1 1 a 4 5 0 1 1 b form. The result will be wherea = ,b = , andc = . 0 0 1 c (b) Use Gauss-Jordan reduction to put the augmented coecient matrix in reduced row 2 3 1 0 0 d 4 5 echelon form. The result will be 0 1 0 e where d = , e = , and 0 0 1 f f = . (c) The solutions of () are x = , y = , and z = . (4) Consider the following system of equations. 0:003000x + 59:14y = 59:17 5:291x 6:130y = 46:78: (a) Using only row operation III and back substitution nd the exact solution of the system. Answer: x = , y = . (b) Same as (a), but after performing each arithmetic operation round o your answer to four signi cant gures. Answer: x = , y = .1.2. EXERCISES 5 (5) Find the values of k for which the system of equations  x +ky = 1 kx +y = 1 has (a) no solution. Answer: . (b) exactly one solution. Answer: . (c) in nitely many solutions. Answer: . (d) When there is exactly one solution, it is x = and y = . (6) Consider the following two systems of equations. 8 x + y + z = 6 x + 2y + 2z = 11 (1) : 2x + 3y 4z = 3 and 8 x + y + z = 7 x + 2y + 2z = 10 (2) : 2x + 3y 4z = 3 Solve both systems simultaneously by applying Gauss-Jordan reduction to an appro- priate 3 5 matrix. 2 3 4 5 (a) The resulting row echelon form of this 3 5 matrix is . 2 3 4 5 (b) The resulting reduced row echelon form is . (c) The solution for (1) is ( , , ) and the solution for (2) is ( , , ) . (7) Consider the following system of equations: 8 x y 3z = 3 2x + z = 0 : 2y + 7z =c (a) For what values of c does the system have a solution? Answer: c = . (b) For the value ofc you found in (a) describe the solution set geometrically as a subset 3 ofR . Answer: . (c) What does part (a) say about the planes xy 3z = 3, 2x +z = 0, and 2y + 7z = 4 3 inR ? Answer: .6 1. SYSTEMS OF LINEAR EQUATIONS (8) Consider the following system of linear equations ( where b ;:::;b are constants). 1 5 8 u + 2v w 2x + 3y =b 1 x y + 2z =b 2 2u + 4v 2w 4x + 7y 4z =b 3 x + y 2z =b 4 : 3u + 6v 3w 6x + 7y + 8z =b 5 (a) In the process of Gaussian elimination the leading variables of this system are and the free variables are . (b) What condition(s) must the constants b ;:::;b satisfy so that the system is consis- 1 5 tent? Answer: . (c) Do the numbers b = 1, b =3, b = 2, b = b = 3 satisfy the condition(s) you 1 2 3 4 5 listed in (b)? . If so, nd the general solution to the system as a function of the free variables. Answer: u = v = w = x = y = z = : (9) Consider the following homogeneous system of linear equations (wherea andb are nonzero constants). 8 x + 2y = 0 ax + 8y + 3z = 0 : by + 5z = 0 (a) Find a value fora which will make it necessary during Gaussian elimination to inter- change rows in the coecient matrix. Answer: a = . (b) Suppose that a does not have the value you found in part (a). Find a value for b so that the system has a nontrivial solution. c d Answer: b = + a where c = and d = . 3 3 (c) Suppose that a does not have the value you found in part (a) and that b = 100. Suppose further that a is chosen so that the solution to the system is not unique.  1 1 The general solution to the system (in terms of the free variable) is z; z; z where = and = .1.3. PROBLEMS 7 1.3. Problems (1) Give a geometric description of a single linear equation in three variables. Then give a geometric description of the solution set of a system of 3 linear equations in 3 variables if the system (a) is inconsistent. (b) is consistent and has no free variables. (c) is consistent and has exactly one free variable. (d) is consistent and has two free variables. (2) Consider the following system of equations:  m x +y =b 1 1 () m x +y =b 2 2 (a) Prove that if m 6=m , then () has exactly one solution. What is it? 1 2 (b) Suppose that m =m . Then under what conditions will () be consistent? 1 2 (c) Restate the results of (a) and (b) in geometrical language.8 1. SYSTEMS OF LINEAR EQUATIONS 1.4. Answers to Odd-Numbered Exercises (1) 2, 3 (3) (a) 2,1, 1 (b) 3,2, 1 (c) 3,2, 1 (5) (a) k =1 (b) k6=1, 1 (c) k = 1 1 1 (d) , k + 1 k + 1 (7) (a)6 (b) a line (c) They have no points in common. (9) (a) 4 (b) 40,10 (c) 10, 20CHAPTER 2 ARITHMETIC OF MATRICES 2.1. Background Topics: addition, scalar multiplication, and multiplication of matrices, inverse of a nonsingular matrix. 2.1.1. De nition. Two square matrices A and B of the same size are said to commute if AB = BA. 2.1.2. De nition. If A and B are square matrices of the same size, then the commutator (or Lie bracket) of A and B, denoted by A;B, is de ned by A;B =ABBA: 2.1.3. Notation. IfA is anmn matrix (that is, a matrix withm rows andn columns), then the th th element in the i row and the j column is denoted by a . The matrix A itself may be denoted ij   m n by a or, more simply, by a . In light of this notation it is reasonable to refer to the ij ij i=1j=1 index i in the expression a as the row index and to call j the column index. When we speak ij th th of the \value of a matrixA at (i;j)," we mean the entry in thei row andj column ofA. Thus, for example, 2 3 1 4 6 7 3 2 6 7 A = 4 5 7 0 5 1 is a 4 2 matrix and a = 7. 31 2.1.4. De nition. A matrix A = a is upper triangular if a = 0 whenever ij. ij ij 2.1.5. De nition. The trace of a square matrix A, denoted by trA, is the sum of the diagonal entries of the matrix. That is, if A = a is an nn matrix, then ij n X trA := a : jj j=1     t 2.1.6. De nition. Thetranspose of annn matrixA = a is the matrixA = a obtained ij ji t by interchanging the rows and columns of A. The matrix A is symmetric if A =A. t t t 2.1.7. Proposition. If A is an mn matrix and B is an np matrix, then (AB) =B A . 910 2. ARITHMETIC OF MATRICES 2.2. Exercises 2 3 2 3 1 0 1 2 1 2   6 7 6 7 0 3 1 1 3 1 3 2 0 5 6 7 6 7 (1) Let A = , B = , and C = . 4 5 4 5 2 4 0 3 0 2 1 0 3 4 3 1 1 2 4 1 (a) Does the matrix D =ABC exist? If so, then d = . 34 (b) Does the matrix E =BAC exist? If so, then e = . 22 (c) Does the matrix F =BCA exist? If so, then f = . 43 (d) Does the matrix G =ACB exist? If so, then g = . 31 (e) Does the matrix H =CAB exist? If so, then h = . 21 (f) Does the matrix J =CBA exist? If so, then j = . 13 "   1 1 1 0 2 2 (2) Let A = , B = , and C =AB. Evaluate the following. 1 1 0 1 2 2 2 3 2 3 6 7 6 7 37 63 6 7 6 7 (a) A = (b) B = 4 5 4 5 2 3 2 3 6 7 6 7 138 42 6 7 6 7 (c) B = (d) C = 4 5 4 5 p Note: If M is a matrix M is the product of p copies of M.   1 1=3 2 (3) Let A = . Find numbers c and d such that A =I. c d Answer: c = and d = . (4) Let A and B be symmetric nn-matrices. Then A;B = B;X, where X = . (5) Let A, B, and C be nn matrices. Then A;BC +BA;C = X;Y , where X = and Y = .   1 1=3 2 (6) Let A = . Find numbers c and d such that A = 0. Answer: c = and c d d = . 2 3 1 3 2 4 5 (7) Consider the matrix a 6 2 where a is a real number. 0 9 5 (a) For what value of a will a row interchange be required during Gaussian elimination? Answer: a = . (b) For what value of a is the matrix singular? Answer: a = . 2 3 2 3 1 0 1 2 1 2   6 7 6 7 0 3 1 1 3 1 3 2 0 5 6 7 6 7 (8) Let A = , B = , C = , and 4 5 4 5 2 4 0 3 0 2 1 0 3 4 3 1 1 2 4 1 3 2 M = 3A 5(BC) . Then m = and m = . 14 41 3 2 (9) If A is an nn matrix and it satis es the equation A 4A + 3A 5I = 0, then A is n nonsingular2.2. EXERCISES 11 and its inverse is . (10) LetA,B, andC benn matrices. Then A;B;C + B;C;A + C;A;B =X, where 2 3 6 7 6 7 X = . 4 5 (11) LetA,B, andC benn matrices. Then A;C + B;C = X;Y , whereX = and Y = . 2 3 2 3 1 0 0 0 6 7 1 6 7 6 7 1 0 0 6 4 7 6 7 (12) Find the inverse of . Answer: . 1 1 4 5 6 7 1 0 3 3 4 5 1 1 1 1 2 2 2 (13) The matrix 2 3 1 1 1 1 2 3 4 1 1 1 1 6 7 6 7 2 3 4 5 H = 6 7 1 1 1 1 4 5 3 4 5 6 1 1 1 1 4 5 6 7 1 is the 44Hilbert matrix. Use Gauss-Jordan elimination to computeK =H . Then 0 K is (exactly) . Now, create a new matrixH by replacing each entry inH 44 1 by its approximation to 3 decimal places. (For example, replace by 0:167.) Use Gauss- 6 0 0 0 Jordan elimination again to nd the inverse K of H . Then K is . 44 (14) Suppose that A and B are symmetric nn matrices. In this exercise we prove that AB is symmetric if and only if A commutes with B. Below are portions of the proof. Fill in the missing steps and the missing reasons. Choose reasons from the following list. (H1) Hypothesis that A and B are symmetric. (H2) Hypothesis that AB is symmetric. (H3) Hypothesis that A commutes with B. (D1) De nition of commutes. (D2) De nition of symmetric. (T) Proposition 2.1.7. Proof. Suppose that AB is symmetric. Then AB = (reason: (H2) and ) t t =B A (reason: ) = (reason: (D2) and ) So A commutes with B (reason: ). Conversely, suppose that A commutes with B. Then t (AB) = (reason: (T) ) =BA (reason: and ) = (reason: and ) Thus AB is symmetric (reason: ). 12 2. ARITHMETIC OF MATRICES 2.3. Problems 2 (1) Let A be a square matrix. Prove that if A is invertible, then so is A. Hint. Our assumption is that there exists a matrix B such that 2 2 A B =BA =I: We want to show that there exists a matrix C such that AC =CA =I: Now to start with, you ought to nd it fairly easy to show that there are matrices L and R such that LA =AR =I: () A matrix L is a left inverse of the matrix A if LA = I; and R is a right inverse of A if AR =I. Thus the problem boils down to determining whether A can have a left inverse and a right inverse that are di erent. (Clearly, if it turns out that they must be the same, then the C we are seeking is their common value.) So try to prove that if () holds, then L =R. (2) Anton speaks French and German; Geraldine speaks English, French and Italian; James speaks English, Italian, and Spanish; Lauren speaks all the languages the others speak   except French; and no one speaks any other language. Make a matrix A = a with ij rows representing the four people mentioned and columns representing the languages they speak. Put a = 1 if person i speaks language j and a = 0 otherwise. Explain the ij ij t t signi cance of the matrices AA and A A. (3) Portland Fast Foods (PFF), which produces 138 food products all made from 87 basic ingredients, wants to set up a simple data structure from which they can quickly extract answers to the following questions: (a) How many ingredients does a given product contain? (b) A given pair of ingredients are used together in how many products? (c) How many ingredients do two given products have in common? (d) In how many products is a given ingredient used? In particular, PFF wants to set up a single table in such a way that: (i) the answer to any of the above questions can be extracted easily and quickly (matrix arithmetic permitted, of course); and (ii) if one of the 87 ingredients is added to or deleted from a product, only a single entry in the table needs to be changed. Is this possible? Explain. (4) Prove proposition 2.1.7. (5) Let A and B be 2 2 matrices. 2 (a) Prove that if the trace of A is 0, then A is a scalar multiple of the identity matrix. (b) Prove that the square of the commutator of A and B commutes with every 2 2 matrix C. Hint. What can you say about the trace of A;B? (c) Prove that the commutator ofA andB can never be a nonzero multiple of the identity matrix.2.3. PROBLEMS 13 (6) The matrices that represent rotations of the xy-plane are   cos sin A() = : sin cos t (a) Let x be the vector (1; 1), = 3=4, and y beA() acting on x (that is, y =A()x ). Make a sketch showing x, y, and . (b) Verify that A( )A( ) =A( + ). Discuss what this means geometrically. 1 2 1 2 (c) What is the product of A() times A()? Discuss what this means geometrically. (d) Two sheets of graph paper are attached at the origin and rotated in such a way that the point (1; 0) on the upper sheet lies directly over the point (5=13; 12=13) on the lower sheet. What point on the lower sheet lies directly below (6; 4) on the upper one? (7) Let 2 3 2 3 4 0 a a a a 2 3 6 7 0 0 a a a 6 7 2 6 7 A = 0 0 0 a a : 6 7 4 5 0 0 0 0 a 0 0 0 0 0 The goal of this problem is to develop a \calculus" for the matrix A. To start, recall 1 (or look up) the power series expansion for . Now see if this formula works for 1x 1 the matrix A by rst computing (IA) directly and then computing the power series expansion substituting A forx. (Explain why there are no convergence diculties for the A series when we use this particular matrix A.) Next try to de ne ln(I +A) and e by ln(I+A) means of appropriate series. Do you get what you expect when you compute e ? Do A A 2A formulas like e e =e hold? What about other familiar properties of the exponential and logarithmic functions? Try some trigonometry withA. Use series to de ne sin, cos, tan, arctan, and so on. Do things like tan(arctan(A)) produce the expected results? Check some of the more obvious 2 2 trigonometric identities. (What do you get for sin A + cos AI? Is cos(2A) the same 2 2 as cos A sin A?) A relationship between the exponential and trigonometric functions is given by the ix famous formula e = cosx +i sinx. Does this hold for A? Do you think there are other matrices for which the same results might hold? Which ones? (8) (a) Give an example of two symmetric matrices whose product is not symmetric. Hint. Matrices containing only 0's and 1's will suce. (b) Now suppose thatA andB are symmetricnn matrices. Prove thatAB is symmetric if and only if A commutes with B. Hint. To prove that a statement P holds \if and only if" a statement Q holds you must rst show that P implies Q and then show that Q implies P. In the current problem, there are 4 conditions to be considered: t (i) A =A (A is symmetric), t (ii) B =B (B is symmetric), t (iii) (AB) =AB (AB is symmetric), and (iv) AB =BA (A commutes with B). Recall also the fact given in (v) theorem 2.1.7. The rst task is to derive (iv) from (i), (ii), (iii), and (v). Then try to derive (iii) from (i), (ii), (iv), and (v).14 2. ARITHMETIC OF MATRICES 2.4. Answers to Odd-Numbered Exercises (1) (a) yes, 142 (b) no, (c) yes,45 (d) no, (e) yes,37 (f) no, (3)6,1 (5) A, BC (7) (a) 2 (b)4 1 2 (9) (A 4A + 3I ) n 5 (11) A +B, C (13) 2800,1329:909CHAPTER 3 ELEMENTARY MATRICES; DETERMINANTS 3.1. Background Topics: elementary (reduction) matrices, determinants. The following de nition says that we often regard the e ect of multiplying a matrix M on the left by another matrix A as the action of A on M. 3.1.1. De nition. We say that the matrix A acts on the matrix M to produce the matrix N if   0 1 N =AM. For example the matrix acts on any 2 2 matrix by interchanging (swapping) 1 0      0 1 a b c d its rows because = . 1 0 c d a b 3.1.2. Notation. We adopt the following notation for elementary matrices which implement type I row operations. LetA be a matrix havingn rows. For any real numberr =6 0 denote byM (r) the j th nn matrix which acts on A by multiplying its j row by r. (See exercise 1.) 3.1.3. Notation. We use the following notation for elementary matrices which implement type II row operations. (See de nition 1.1.1.) Let A be a matrix having n rows. Denote by P the nn ij th th matrix which acts on A by interchanging its i and j rows. (See exercise 2.) 3.1.4. Notation. And we use the following notation for elementary matrices which implement type III row operations. (See de nition 1.1.1.) Let A be a matrix having n rows. For any real th number r denote by E (r) the nn matrix which acts on A by adding r times the j row of A ij th to the i row. (See exercise 3.) 3.1.5. De nition. If a matrix B can be produced from a matrix A by a sequence of elementary row operations, then A and B are row equivalent. Some Facts about Determinants 3.1.6. Proposition. Let n2N and M be the collection of all nn matrices. There is exactly nn one function det: M R: A7 detA nn which satis es (a) detI = 1. n 0 0 (b) If A2 M and A is the matrix obtained by interchanging two rows of A, then detA = nn detA. 0 (c) If A2 M , c2 R, and A is the matrix obtained by multiplying each element in one nn 0 row of A by the number c, then detA =c detA. 0 (d) If A2 M , c2R, and A is the matrix obtained from A by multiplying one row of A nn by c and adding it to another row of A (that is, choose i and j between 1 and n with i6=j 0 and replace a by a +ca for 1kn), then detA = detA. jk jk ik 1516 3. ELEMENTARY MATRICES; DETERMINANTS 3.1.7. De nition. The unique function det: M R described above is the nn determi- nn nant function. 3.1.8. Proposition. If A = a for a2R (that is, if A2 M ), then detA = a; if A2 M , 11 22 then detA =a a a a : 11 22 12 21 3.1.9. Proposition. If A;B2 M , then det(AB) = (detA)(detB). nn t 3.1.10. Proposition. If A 2 M , then detA = detA. (An obvious corollary of this: in nn conditions (b), (c), and (d) of proposition 3.1.6 the word \columns" may be substituted for the word \rows".) 3.1.11. De nition. LetA be annn matrix. Theminor of the elementa , denoted byM , is jk jk th the determinant of the (n 1) (n 1) matrix which results from the deletion of the j row and th k column of A. The cofactor of the element a , denoted by C is de ned by jk jk j+k C := (1) M : jk jk 3.1.12. Proposition. If A2 M and 1jn, then nn n X detA = a C : jk jk k=1 th This is the (Laplace) expansion of the determinant along the j row. In light of 3.1.10, it is clear that expansion along columns works as well as expansion along rows. That is, n X detA = a C jk jk j=1 th for any k between 1 and n. This is the (Laplace) expansion of the determinant along the k column. 3.1.13. Proposition. An nn matrix A is invertible if and only if detA =6 0. If A is invertible, then 1 1 t A = (detA) C   where C = C is the matrix of cofactors of elements of A. jk3.2. EXERCISES 17 3.2. Exercises rd (1) LetA be a matrix with 4 rows. The matrix M (4) which multiplies the 3 row ofA by 4 3 2 3 6 7 6 7 is . (See 3.1.2.) 4 5 nd th (2) Let A be a matrix with 4 rows. The matrix P which interchanges the 2 and 4 rows 24 2 3 6 7 6 7 of A is . (See 3.1.3.) 4 5 rd (3) Let A be a matrix with 4 rows. The matrix E (2) which adds2 times the 3 row of 23 2 3 6 7 nd 6 7 A to the 2 row is . (See 3.1.4.) 4 5 2 3 6 7 11 6 7 (4) Let A be the 4 4 elementary matrix E (6). Then A = and 43 2 3 4 5 6 7 9 6 7 A = . 4 5 2 3 6 7 9 6 7 (5) Let B be the elementary 4 4 matrix P . Then B = and 24 2 3 4 5 6 7 10 6 7 B = . 4 5 2 3 6 7 4 6 7 (6) Let C be the elementary 4 4 matrix M (2). Then C = and 3 2 3 4 5 6 7 3 6 7 C = . 4 5 2 3 1 2 3 6 7 0 1 1 6 7 (7) Let A = and B = P E (2)M (2)E (1)P A. Then b = 23 34 3 42 14 23 4 5 2 1 0 1 2 3 and b = . 32 (8) We apply Gaussian elimination (using type III elementary row operations only) to put a 4 4 matrix A into upper triangular form. The result is 2 3 1 2 2 0 6 7 0 1 0 1 5 6 7 E ( )E (2)E (1)E (2)A = : 43 42 31 21 2 4 5 0 0 2 2 0 0 0 10 Then the determinant of A is .18 3. ELEMENTARY MATRICES; DETERMINANTS (9) The system of equations: 8 2y+3z = 7 x+ y z =2 : x+ y5z = 0 is solved by applying Gauss-Jordan reduction to the augmented coecient matrix 2 3 0 2 3 7 4 5 A = 1 1 1 2 . Give the names of the elementary 3 3 matrices X ;:::;X 1 8 1 1 5 0 which implement the following reduction. 2 3 2 3 2 3 1 1 1 2 1 1 1 2 1 1 1 2 X X X 1 2 3 4 5 4 5 4 5 A 0 2 3 7 0 2 3 7 0 2 3 7 1 1 5 0 0 2 6 2 0 0 9 9 2 3 2 3 2 3 1 1 1 2 1 1 1 2 1 1 1 2 X X X 4 5 6 4 5 4 5 4 5 0 2 3 7 0 2 0 4 0 1 0 2 0 0 1 1 0 0 1 1 0 0 1 1 2 3 2 3 1 1 0 1 1 0 0 3 X X 7 8 4 5 4 5 0 1 0 2 0 1 0 2 : 0 0 1 1 0 0 1 1 Answer: X = , X = , X = , X = , 1 2 3 4 X = , X = , X = , X = . 5 6 7 8 (10) Solve the following equation for x: 2 3 3 4 7 0 6 2 6 7 2 0 1 8 0 0 6 7 6 7 3 4 8 3 1 2 6 7 det = 0: Answer: x = . 6 7 27 6 5 0 0 3 6 7 4 5 3 x 0 2 1 1 1 0 1 3 4 0 2 3 0 0 1 1 4 5 0 2 4 (11) LetA = . FindA using the technique of augmentingA by the identity matrix 1 2 3 I and performing Gauss-Jordan reduction on the augmented matrix. The reduction can be accomplished by the application of ve elementary 3 3 matrices. Find elementary 1 matrices X , X , and X such that A =X E (3)X M (1=2)X I. 1 2 3 3 13 2 2 1 (a) The required matrices areX =P wherei = ,X =E (2) wherej = 1 1i 2 jk and k = , and X = E (r) where r = . 3 12 2 3 6 7 1 6 7 (b) And then A = . 4 5 2 3 2 3 1 t t t 2 6 7 t 1 t t p 6 7 (12) det = (1a(t)) where a(t) = and p = . 2 4 5 t t 1 t 3 2 t t t 13.2. EXERCISES 19 (13) Evaluate each of the following determinants. 2 3 6 9 39 49 6 7 5 7 32 37 6 7 (a) det = . 4 5 3 4 4 4 1 1 1 1 2 3 1 0 1 1 6 7 1 1 2 0 6 7 (b) det = . 4 5 2 1 3 1 4 17 0 5 2 3 13 3 8 6 6 7 0 0 4 0 6 7 (c) det = . 4 5 1 0 7 2 3 0 2 0 2 3 5 4 2 3 6 7 5 7 1 8 6 7 (14) Let M be the matrix . 4 5 5 7 6 10 5 7 1 9 (a) The determinant of M can be expressed as the constant 5 times the determinant of 2 3 3 1 5 4 5 3 the single 3 3 matrix . 3 (b) The determinant of this 3 3 matrix can be expressed as the constant 3 times the   7 2 determinant of the single 2 2 matrix . 2 (c) The determinant of this 2 2 matrix is . (d) Thus the determinant of M is . 2 3 1 2 5 7 10 6 7 1 2 3 6 7 6 7 6 7 (15) Find the determinant of the matrix 1 1 3 5 5 . Answer: . 6 7 4 5 1 1 2 4 5 1 1 1 1 1 (16) Find the determinants of the following matrices. 2 3 2 3 73 78 24 73 78 24 4 5 4 5 A = 92 66 25 and B = 92 66 25 : 80 37 10 80 37 10:01 Hint. Use a calculator (thoughtfully). Answer: detA = and detB = . (17) Find the determinant of the following matrix. 2 3 83 15 283 5  347:86 10 6 7 3136 56 5 cos(2:7402) 6 7 : 4 5 6776 121 11 5 2464 44 4 2 Hint. Do not use a calculator. Answer: .20 3. ELEMENTARY MATRICES; DETERMINANTS 2 3 1 1 0 0 2 2 6 1 17 0 0 6 7 2 2 1 (18) LetA =6 7. We ndA using elementary row operations to convert the 1 1 0 0 4 5 2 2 1 1 1 0 2 2 h i h i . . . . 1 4 8 matrix to the matrix . A . I I . A 4 4 Give the names of the elementary 4 4 matrices X ;:::;X which implement the 1 11 following Gauss-Jordan reduction and ll in the missing matrix entries. 2 3 2 3 . . 1 1 1 1 . . 0 0 . 1 0 0 0 1 0 . 2 2 2 2 6 7 6 7 . . 6 7 6 7 1 1 1 1 . . X 60 0 . 0 1 0 07 1 60 0 . 7 2 2 2 2 // 6 7 6 7 . . 6 7 6 7 1 1 1 1 . . 0 0 . 0 0 1 0 0 0 . 4 5 4 5 2 2 2 2 . . 1 1 1 1 . . 1 0 . 0 0 0 1 0 0 . 2 2 2 2 2 3 2 3 . . 1 1 . 1 1 . 1 0 . 1 0 . 2 2 2 2 6 7 6 7 . . 6 7 6 7 1 1 . 1 1 . X X 0 0 . 0 0 . 2 6 7 3 6 7 // 2 2// 2 2 6 7 6 7 . . 6 7 6 7 3 1 . 3 1 . 0 0 . 0 0 . 4 5 4 5 4 4 4 4 . . 1 1 . 1 1 . 0 0 . 0 0 . 2 2 2 2 2 3 2 3 . . 1 1 . 1 1 . 1 0 . 1 0 . 2 2 2 2 6 7 6 7 . . 6 7 6 7 . . X 0 1 0 1 . X 0 1 0 1 . 4 6 7 5 6 7 //// 6 7 6 7 . . 6 7 6 7 3 1 . 1 . 0 0 . 0 0 1 . 4 5 4 5 4 4 3 . . 1 1 . 1 1 . 0 0 . 0 0 . 2 2 2 2 2 3 2 3 . . 1 1 . 1 1 . 1 0 . 1 0 . 2 2 2 2 6 7 6 7 . . 6 7 6 7 . . X 0 1 0 1 . X 0 1 0 1 . 6 6 7 7 6 7 //// 6 7 6 7 . . 6 7 6 7 1 . . 0 0 1 . 0 0 1 0 . 4 5 4 5 3 . . 1 . 1 . 0 0 0 . 0 0 0 . 3 3 2 3 2 3 . . 1 1 . 1 1 . 1 0 . 1 0 . 2 2 2 2 6 7 6 7 . . 6 7 6 7 . . X 0 1 0 1 . X 0 1 0 0 . 8 6 7 9 6 7 //// 6 7 6 7 . . 6 7 6 7 . . 0 0 1 0 . 0 0 1 0 . 4 5 4 5 . . . . 0 0 0 1 . 0 0 0 1 . 2 3 2 3 . . 1 . . 1 0 0 . 1 0 0 0 . 2 6 7 6 7 . . 6 7 6 7 . . X X 10 60 1 0 0 . 7 11 60 1 0 0 . 7 //// 6 7 6 7 . . 6 7 6 7 . . 0 0 1 0 . 0 0 1 0 . 4 5 4 5 . . . . 0 0 0 1 . 0 0 0 1 . Answer: X = , X = , X = , X = , 1 2 3 4 X = , X = , X = , X = . 5 6 7 8 X = , X = , X = . 9 10 113.2. EXERCISES 21 (19) Suppose that A is a square matrix with determinant 7. Then (a) det(P A) = . 24 (b) det(E (4)A) = . 23 (c) det(M (2)A) = . 3

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