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linear algebra

Linear Algebra A Modern Introduction 4th edition David Poole - Solutions

For Exercises 1 and 2, determine the currents for the given electrical networks.1.2.
Suppose that in Example 2.27, 400 units of food A, 500 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.7. How many bacteria of each strain can coexist in the test tube and consume
(a) Find the currents I, I1( ( ( ( ( I5 in the bridge circuit in Figure 2.22.(b) Find the effective resistance of this network.(c) Can you change the resistance in branch BC (but leave everything else unchanged) so that the current through branch CE becomes
The networks in parts (a) and (b) of Figure 2.23 show two resistors coupled in series and in parallel, respectively. We wish to find a general formula for the effective resistance of each network-that is, find Reff such that E = Reffl.(a) Show that the effective resistance Reff of a network with
Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs
Suppose the coal and steel industries form a closed economy. Every $1 produced by the coal industry requires $0.30 of coal and $0.70 of steel. Every $ 1 produced by steel requires $0.80 o f coal and $0.20 of steel. Find the annual production (output) of coal and steel if the total annual production
A painter, a plumber, and an electrician enter into a cooperative arrangement in which each of them agrees to work for himself/herself and the other two for a total of 10 hours per week according to the schedule shown in Table 2.8. For tax purposes, each person must establish a value for his/her
Four neighbors, each with a vegetable garden, agree to share their produce. One will grow beans (B), one will grow lettuce (L), one will grow tomatoes (T), and one will grow zucchini (Z). Table 2.9 shows what fraction of each crop each neighbor will receive. What prices should the neighbors charge
Suppose the coal and steel industries form an open economy. Every $1 produced by the coal industry requires $0.15 of coal and $0.20 of steel. Every $ 1 produced by steel requires $0.25 of coal and $0.10 of steel. Suppose that there is an annual outside demand for $45 million of coal and $ 1 24
In Gotham City, the departments of Administration (A), Health (H), and Transportation (T) are interdependent. For every dollar's worth of services they produce, each department uses a certain amount of the services produced by the other departments and itself, as shown in Table 2.10. Suppose that,
(a) In Example 2.35, suppose all the lights are initially off. Can we push the switches in some order so that only the second and fourth lights will be on?(b) Can we push the switches in some order so that only the second light will be on?
A florist offers three sizes of flower arrangements containing roses, daisies, and chrysanthemums. Each small arrangement contains one rose, three daisies, and three chrysanthemums. Each medium arrangement contains two roses, four daisies, and six chrysanthemums. Each large arrangement contains
(a) In Example 2.35, suppose the fourth light is initially on and the other four lights are off. Can we push the switches in some order so that only the second and fourth lights will be on?(b) Can we push the switches in some order so that only the second light will be on?
In Example 2.35, describe all possible configurations of lights that can be obtained if we start with all the lights off.
(a) In Example 2.36, suppose that all of the lights are initially off. Show that it is possible to push the switches in some order so that the lights are off, dark blue, and light blue, in that order.(b) Show that it is possible to push the switches in some order so that the lights are light blue,
Suppose the lights in Example 2.35 can be off, light blue, or dark blue and the switches work as described in Example 2.36. (That is, the switches control the same lights as in Example 2.35 but cycle through the colors as in Example 2.36.) Show that it is possible to start with all of the lights
For Exercise 33, describe all possible configurations of lights that can be obtained, starting with all the lights off.
Nine squares, each one either black or white, are arranged in a 3 X 3 grid. Figure 2.24 shows one possibleFigure 2.24The nine squares puzzleArrangement. When touched, each square changes its own state and the states of some of its neighbors (black ( white and white ( black). Figure 2.25 showsFigure
Consider a variation on the nine squares puzzle. The game is the same as that described in Exercise 35 except that there are three possible states for each square: white, gray, or black. The squares change as shown in Figure 2.25, but now the state changes follow the cycle white ( gray ( black (
In Exercises 1-3, set up and solve an appropriate system of linear equations to answer the questions. 1. Grace is three times as old as Hans, but in 5 years she will be twice as old as Hans is then. How old are they now? 2. The sum of Annie's, Bert's, and Chris's ages is 60. Annie is older than
(a) In your pocket you have some nickels, dimes, and quarters. There are 20 coins altogether and exactly twice as many dimes as nickels. The total value of the coins is $3.00. Find the number of coins of each type. (b) Find all possible combinations of 20 coins (nickels, dimes, and quarters) that
A coffee merchant sells three blends of coffee. A bag of the house blend contains 300 grams of Colombian beans and 200 grams of French roast beans. A bag of the special blend contains 200 grams of Colombian beans, 200 grams of Kenyan beans, and 100 grams of French roast beans. A bag of the gourmet
Redo Exercise 5, assuming that the house blend contains 300 grams of Colombian beans, 50 grams of Kenyan beans, and 1 50 grams of French roast beans and the gourmet blend contains 100 grams of Colombian beans, 350 grams of Kenyan beans, and 50 grams of French roast beans. This time the merchant has
In Exercises 1-3, balance the chemical equation for each reaction.
In Exercises 1-3, apply Jacobi's method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each variable. In each case, compare your answer with the exact solution found using any
In Exercises 1 and 2, compute the first four iterates, using the zero vector as the initial approximation, to show that the Gauss-Seidel method diverges. Then show that the equations can be rearranged to give a strictly diagonally dominant coefficient matrix, and apply the Gauss-Seidel method to
Draw a diagram to illustrate the divergence of the Gauss-Seidel method in Exercise 15.
In Exercises 1 and 2, the coefficient matrix is not strictly diagonally dominant, nor can the equations be rearranged to make it so. However, both the Jacobi and the Gauss-Seidel method converge anyway. Demonstrate that this is true of the Gauss-Seidel method, starting with the zero vector as the
Continue performing iterations in Exercise 18 to obtain a solution that is accurate to within 0.00 1.
Continue performing iterations in Exercise 19 to obtain a solution that is accurate to within 0.00 1.
In Exercises 22-24, the metal plate has the constant temperatures shown on its boundaries. Find the equilibrium temperature at each of the indicated interior points by setting up a system of linear equations and applying either the Jacobi or the Gauss-Seidel method. Obtain a solution that is
In Exercises 1 and 2, we refine the grids used in Exercises 22 and 24 to obtain more accurate information about the equilibrium temperatures at interior points of the plates. Obtain solutions that are accurate to within 0.001, using either the Jacobi or the Gauss-Seidel method. 1. 2.
Exercises 1 and 2 demonstrate that sometimes, if we are lucky, the form of an iterative problem may allow us to use a little insight to obtain an exact solution. 1. A narrow strip of paper 1 unit long is placed along a number line so that its ends are at 0 and 1. The paper is folded in half, right
Mark each of the following statements true or false:
Find the general equation of the plane spanned by
Determine whetherAre linearly independent.
(a)
(a) The reduced row echelon form of A is I3.(b) The rank of A is 3.
Let a1, a2, a3 be linearly dependent vectors in, not all zero, and let A = [a1 a2 a3]. What are the possible values of the rank of A?
What is the maximum rank of a 5 ( 3 matrix? What is the minimum rank of a 5 ( 3 matrix?
Show that if u and v are linearly independent vectors, then so are u + v and u - v.
Show that span (u, v) = span (u, u + v) for any vectors u and v.
In order for a linear system with augmented matrix [A ( b] to be consistent, what must be true about the ranks of A and [A ( b]?
Find the rank of the matrix
Are the matricesRow equivalent? Why or why not?
Solve the linear system x + y - 2z = 4 x + 3y - z = 7 2x + y - 5z = 7
Solve the linear system 3w + 8x - 18y + z = 35 w + 2x - 4y = 11 w + 3x - 7y + z = 10
Solve the linear system2x + 3y = 4x + 2y = 3
Solve the linear system3x + 2y = 1x + 4y = 2
For what value(s) of k is the linear system with augmented matrixinconsistent?
Find parametric equations for the line of intersection of the planes x + 2y + 3z = 4 and Sx + 6y + 7z = 8.
Find the point of intersection of the following lines, if it exists.
In Exercises, compute the indicated matrices (if possible).Let1. A + 2D 2. 3D - 2A 3. B - C
Give an example of a nonzero 2 × 2 matrix A such that A2 = O.
A factory manufactures three products (doohickies, gizmos, and widgets) and ships them to two warehouses for storage. The number of units of each product shipped to each warehouse is given by the matrix(Where aij is the number of units of product i sent to warehouse j and the products are taken in
Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the same for each product but varies by warehouse because of the distances involved. It costs $0.75 to distribute one unit from warehouse 1 and $1 .00 to distribute one unit from warehouse 2. Organize
In Exercises, write the given system of linear equations as a matrix equation of the form Ax = b. a. x1 - 2x2 + 3x3 = 0 2x1 + x2 - 5x3 = 4 b. -x1 + 2x3 = 1 x1 - x2 = - 2 x2 + x3 = - 1
In Exercises, assume that the product AB makes sense. a. Prove that if the columns of B are linearly dependent, then so are the columns of AB. b. Prove that if the rows of A are linearly dependent, then so are the rows of AB.
In Exercises, compute AB by block multiplication, using the indicated partitioninga.b.
In each of the following, find the 4 × 4 matrix A = [ aij] that satisfies the given condition:(a) aij = ( - l)i+j(b) aij = j - i(c) aij = (i - l)j (d) sin ((i + j - 1 )π) / 4)
In each of the following, find the 6 Ã— 6 matrix A = [aij] that satisfies the given condition:a.b. c.
In Exercises, solve the equation for X, given thata. X -2A + 3B = 0 b. 2X = A -B c. 2(A + 2B) = 3X
In Exercises, determine whether the given matrices are linearly independent.a.b. c.
Prove that, for square matrices A and B, AB = BA if and only if (A - B) (A + B) = A2 - B2
In Exercises, ifFind conditions on a, b, c, and d such that AB = BA. a. b. c.
Find conditions on a, b, c, and d such thatCommutes with both And
Find conditions on a, b, c, and d such thatCommutes with every 2 Ã— 2 matrix
Prove that if AB and BA are both defined, then AB and BA are both square matrices.
A square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix isWhere the entries marked * are arbitrary. A more formal definition of such a matrix A=[aij] is that aij = 0 if i > j. Prove that the product of two upper
Using induction, prove that for all n ≥ 1, (A1 + A2 + · · · + An)T = ATI + AT2 + · · · + ATn.
Using induction, prove that for all n ≥ 1, (A1 A1· · · An)T = ATn· · · AT2AT1.
(a) Prove that if A and B are symmetric n × n matrices, then so is A + B. (b) Prove that if A is a symmetric n × n matrix, then so is kA for any scalar k.
(a) Give an example to show that if A and B are symmetric n × n matrices, then AB need not be symmetric. (b) Prove that if A and B are symmetric n × n matrices, then AB is symmetric if and only if AB = BA.
A square matrix is called skew-symmetric if AT = -A.Which of the following matrices are skew-symmetric?a.b.c.d.
Give a component wise definition of a skew-symmetric matrix.
Prove that if A and B are skew-symmetric n × n matrices, then so is A + B.
If A and B are skew-symmetric 2 × 2 matrices, under what conditions is AB skew-symmetric?
Prove that if A is an n × n matrix, then A - AT is skew-symmetric.
(a) Prove that any square matrix A can be written as the sum of a symmetric matrix and a skew symmetric matrix.(b) Illustrate pact (a) for the matrix
The trace of an n X n matrix A = [aij] is the sum of the entries on its main diagonal and is denoted by tr(A). That is,tr(A) = a11 + a22 + · · · + ann If A and B are n × n matrices, prove the following properties of the trace:(a) tr (A + B) = tr (A) + tr (B)(b) tr (kA) = ktr (A), where k is a
Prove that if A and B are n × n matrices, then tr (AB) = tr (BA).
If A is any matrix, to what is tr (AAT) equal?
Show that there are no 2 × 2 matrices A and B such that AB - BA = 12.
In Exercises, write B as a linear combination of the other matrices, if possible.a.b. c.
In Exercises, find the general form of the span of the indicated matrices.a. Span (A1, A2) in Exercise 5 In Exercise 5 b. Span (A1, A2, A3) in Exercise 6 In Exercise 6
In Exercises, find the inverse of the given matrix (if it exists) using Theorem 3.8.a.b. c.
In Exercises, solve the given system using the method of Example 3.25. a. 2x + y = - 1 5x + 3y = 2 b. x1 - x2 = 1 2x1 + x2 = 2
(a) Find A-1 and use it to solve the three systems Ax = b1, Ax = b2, and Ax = h3.(b) Solve all three systems at the same time by row reducing the augmented matrix [A| b1 b2 b3] using Gauss-Jordan elimination(c) Carefully count the total number of individual multiplications that you performed in (a)
Prove that the n × n identity matrix In is invertible and that ln-l = In
(a) Give a counterexample to show that (AB)-1 ≠ A-1 B-1 in general. (b) Under what conditions on A and B is (AB)-1 = A-1 B-1? Prove your assertion.
By induction, prove that if A1, A2. . . An are invertible matrices of the same size, then the product A1A2 · · · An is invertible and (A1A2 . . . An)-1 = A =An-1 · · · A2-1A1-1.
Give a counterexample to show that (A + B)-1 ≠ A-1 + B-1 in general.
In Exercises, solve the given matrix equation for X. Simplify your answers as much as possible. (In the words of Albert Einstein, "Everything should be made as simple as possible, but not simpler.") Assume that all matrices are invertible. a. XA2 = A-1 b. AXB = (BA)2 c. (A-1 X)-1 = A(B-2 A)-1 d.
In Exercises, find the inverse of the given elementary matrix.a.b. c. d.
In Exercises, find a sequence of elementary matrices E1, E2, . . , Ek such that Ek · · · E2E1A = I. Use this sequence to write both A and A-1 as products of elementary matrices.a.b.
(a) Prove that if A is invertible and AB = 0, then B = 0. (b) Give a counterexample to show that the result in part a may fail if A is not invertible.
(a) Prove that if A is invertible and BA = CA, then B = C. (b) Give a counterexample to show that the result in part a may fail if A is not invertible.
A square matrix A is called idempotent if A2 = A. (The word idempotent comes from the Latin idem, meaning "same;' and potere, meaning "to have power:' thus, something that is idempotent has the "same power" when squared.) (a) Find three idempotent 2 × 2 matrices. (b) Prove that the only invertible
Show that if A is a square matrix that satisfies the equation A2 - 2A+I=0, then A-1=2I - A.
Prove that if a symmetric matrix is invertible, then its inverse is symmetric also.

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