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modern engineering mathematics

Modern Engineering Mathematics 6th Edition Glyn James - Solutions

Solve the equationsusing the tridiagonal algorithm.
Solve the equationsusing Gaussian elimination.
Solve, using Gaussian elimination with partial pivoting, the following equations:
The two almost identical matrix equations are givenUse MATLAB or MAPLE to show that the solutions are wildly different. Evaluate the determinants of the two 3 × 3 matrices.
Show that a tridiagonal matrix can be written in the formA matrix that has zeros in every position below the diagonal is called an upper-triangular matrix and one with zeros everywhere above the diagonal is called a lower-triangular matrix. A matrix that only has non-zero elements in certain
A wire is loaded with equal weights W at nine uniformly spaced points, as illustrated in Figure 5.13. The wire is sufficiently taut that the tension T may be considered to be constant. The end points are at the same level, so that u0 = u10 = 0 and the system is symmetrical about its midpoint. The
Solve the equations in Question 73 (Exercises 5.5.3) using Jacobi iteration starting from the estimate X = [1 1 1 1]T. How accurate is the solution obtained after five iterations?Data from question 73Solve the equationsusing the tridiagonal algorithm.
Solve the equations in Question 74 (Exercises 5.5.3) using Gauss–Seidel iteration, starting from the estimate X = [1 0 0 0]T. How accurate is the solution obtained after three iterations?Data from Question 74Solve the equationsusing Gaussian elimination.
Write a computer program in MATLAB or similar package to obtain the solution, by SOR, to the equations in Question 75 (Exercises 5.5.3). Determine the optimum SOR factor for each equation.Data from Question 75Solve, using Gaussian elimination with partial pivoting, the following equations:
Use an SOR program to solve the equationsso that successive iterations differ by no more than 1 in the fourth decimal place. Find an SOR factor that produces this convergence in less than fifty iterations.
Use an SOR program to solve the equationsso that successive iterations differ by no more than 1 in the fourth decimal place. Find an SOR factor that produces this convergence in less than fifty iterations.
Use an SOR program to solve the equationsso that successive iterations differ by no more than 1 in the fourth decimal place. Find an SOR factor that produces this convergence in less than fifty iterations.
Show that the circuit in Figure 5.18 has equationsTake R1 = 1, R2 = 2, R3 = 2, R4 = 2 and R5 = 3 (all in Ω) and E = 1.5 V. Show that the equations are diagonally dominant, and hence solve the equations by an iterative method.Figure 5.18
Solve the 10 × 10 matrix equation in Example 5.30 using an iterative method starting from X = [1 1 1 1 1 1 1 1 1 1]T. Verify that a solution to four-figure accuracy can be obtained in less than ten iterations.Data from Example 5.30Solve the matrix equation AX = c where
Find the rank of A and of the augmented matrix (A : b). Solve AX = b where possible and check that there are (n – rank(A)) free parameters.
Find the rank of the coefficient matrix and of the augmented matrix in the matrix equationFor each value of α, find, where possible, the solution of the equation.
Find the rank of the matrices
Reduce the matrices in the following equations to echelon form, determine their ranks and solve the equations, if a solution exists:
By obtaining the order of the largest square submatrix with non-zero determinant, determine the rank of the matrixReduce the matrix to echelon form and confirm your result. Check the rank of the augmented matrix (A : b), where bT = [–1 0 –1 0]. Does the equation AX = b have a solution?
Solve, where possible, the following matrix equations:
In a fluid flow problem there are five natural parameters. These have dimensions in terms of length L, mass M and time T as follows:To determine how many non-dimensional parameters can be constructed, seek values of p, q, r, s and t so that VprqDrgsmt is dimensionless. Write the equations for p, q,
Four points in a three-dimensional space have coordinates (xi, yi, zi) for i = 1, . . . , 4. From the rank of the matrixdetermine whether the points lie on a plane or a line or whether there are other possibilities.
A popular method of numerical integration – involves Gaussian integration; it is used in finite-element calculations which are well used in most of engineering. As a simple example, the numerical integral over the interval –1 ≤ x ≤ 1 is writtenUse Gaussian elimination to reduce the
Obtain the characteristic polynomials of the matricesand hence evaluate the eigenvalues of the matrices.
Find the eigenvalues and corresponding eigenvectors of the matrices
Find the eigenvalues and eigenvectors of the matrices
Show that the matrixhas eigenvalues 5, –3, –3. Find the corresponding eigenvectors. For the repeated eigenvalue, show that it has two linearly independent eigenvectors and that any vector of the general formis also an eigenvector.
Obtain the eigenvalues and corresponding eigenvectors of the matrices
Given that λ = 1 is a three-times repeated eigenvalue of the matrixdetermine how many independent eigenvectors correspond to this value of λ. Determine a corresponding set of independent eigenvectors.
Given that λ = 1 is a twice-repeated eigenvalue of the matrixdetermine a set of independent eigenvectors.
Find all the eigenvalues and eigenvectors of the matrix
Verify Properties 1–8 of Section 5.7.6.Data from Section 5.7.6.
Given that the eigenvalues of the matrixare 5, 3 and 1:(a) Confirm Properties 1–4 of Section 5.7.6;(b) Taking k = 2, confirm Properties 5–8 of Section 5.7.6.Data from Section 5.7.6.
Determine the eigenvalues and corresponding eigenvectors of the symmetric matrixand verify that the eigenvectors are mutually orthogonal.
The 3 × 3 symmetric matrix A has eigenvalues 6, 3 and 2. The eigenvectors corresponding to the eigenvalues 6 and 3 are [1 1 2]T and [1 1 –1]T respectively. Find an eigenvector corresponding to the eigenvalue 2.
Verify that the matrixhas eigenvalues ± 1/4 and corresponding eigenvectorsWhat are the eigenvalues of An? Show that any vectorcan be written as Z = αX + βY and hence deduce that
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6}, B = {1, 3, 5, 7} and C = {2, 3, 4, 7, 8} find the sets
LetWrite down the sets(a) A ∩ B(b) A ∩ B ∩ C(c) A ∪ (B ∩ C) and verify that (A ∪ B) ∩ (A ∪ C) = A ∪ (B ∩ C).
If A, B and C are defined as in Question 2, and the universal set is the set of all integers less than or equal to 20, find the following sets:Verify the De Morgan laws for A and B.Data from Question 2LetWrite down the sets(a) A ∩ B(b) A ∩ B ∩ C(c) A ∪ (B ∩ C)and verify that (A ∪ B)
The sets A and B are defined byWhich of the following statements is true?(a) A ≠ B(b) A = B Give reasons for your answers.
(a) Simplify the Boolean functions(b) Draw Venn diagrams to verify that
In an election there are three candidates and 800 voters. The voters may exercise one, two or three votes each. The following results were obtained:Show that these results are inconsistent if all the voters use at least one vote.
Draw switching circuits to establish the truth of the following laws:Use these to simplify the expressionso that s only contains two pairs of products added.
Write down, in set theory notation, expressions corresponding to the outputs in(a) Figure 6.53(b) Figure 6.54.
Draw a switching circuit with inputs x, y, z and u to correspond to the following expressions:For (c) establish the output for the input states(i) x = y = 1, z = u = 0(ii) x = 1, y = z = u = 0
Write down truth tables for the following expressions:The contrapositive of the conditional statement p → q is defined as q̂ → p̂.(d) Use truth tables to show that(e) Use truth tables to evaluate the status of the expression(f) By taking the contrapositive of this conditional statement and
Reduce the following Boolean expressions by taking complements:
(a) Simplify the Boolean expressions(b) Show the Boolean functionon a Venn diagram.
A lift (elevator) services three floors. On each floor there is a call button to call the lift. It is assumed that at the moment of call the cabin is stationary at one of the three floors. Using these six input variables, determine a control that moves the motor in the right direction for the
There are four people on a TV game show. Each has a ‘Yes/No’ button for recording opinions. The display must register ‘Yes’ or ‘No’ according to a majority vote.(a) Derive a truth table for the above.(b) Write down the Boolean expression for the output.(c) Simplify this expression and
Consider the following logical statements:(a) Mike never smokes dope.(b) Rick smokes if, and only if, Mike and Vivian are present.(c) Neil smokes under all conditions – even by himself.(d) Vivian smokes if, and only if, Mike is not present.The police raid: determine the state of there being no
Find the explicit Boolean function for the logic circuit of Figure 6.55. Show that the function simplifies to f = q · r̂ and draw two different simplified circuits which may be used to represent the circuit.
Which of the following statements are propositions? For those that are not, say why and suggest ways of changing them so that they become propositions. For those that are, comment on their truth value.(a) Julius Caesar was prime minister of Great Britain.(b) Stop hitting me.(c) Turn right at the
(a) Draw up truth tables to represent the statements (i) p is equivalent to q (ii) p implies q(b) Using the algebra of statements, represent the truth of the statements below in tabular form and hence determine whether they are true or false:(i) If p implies q, and r implies q, then either r
A panel light in the control room of a satellite launching site is to go on if the pressure in both the oxidizer and fuel tanks is equal to or above a required minimum value and there are 15 minutes or less to ‘lift-off ’, or if the pressure in the oxidizer tank is equal to or above the
In the control problem show that h may also be expressed asCompare the resulting control switching circuit with that of Figure 6.52.Figure 6.52.
Write down all subsets of the set A = { p, q, r, s} that contain the product of four of p, q, r, s or their complement. Represent these on a Venn diagram. [The ideas are pursued through Karnaugh maps which are outside the scope of this text.]Figure 6.52
State the converse and contrapositive of each of the following statements:(a) If the train is late, I will not go.(b) If you have enough money, you will retire.(c) I cannot do it unless you are there too.(d) If you go, so will I.
An island is inhabited by two tribes of vicious cannibals and, sadly, you are a prisoner of one of them. One tribe always tell the truth, the other tribe always lie. Unfortunately both tribes look identical. They will answer ‘yes’ or ‘no’ to a single question they will allow you. The God of
LetFind, where possible,(a) A + B,(b) A + C,(c) C – A,(d) 3A,(e) 4B,(f) C + B,(g) 3A + 2C,(h) AT + A(i) A + CT + BT.
A local roadside cafe serves beefburgers, eggs, chips and beans in four combination meals:A party orders 1 slimmer, 4 normal, 2 jumbo and 2 veggie meals. What is the total amount of materials that the kitchen staff need to cook? One of the customers sees the size of a jumbo meal and changes his
(a) Show that the only solution to the vector equationis α = b = γ = 0.(b) Find a non-zero solution for α, β, γ, δ to the vector equation
Givenfind(a) AB,(b) BA,(c) Bb,(d) ATb,(e) cT(ATb)(f) AC.
Ifevaluate(a) XTX,(b) AX,(c) XT(AX) and(d) 1/2XT[(AT + A)X].
Given the matrices(a) Find(i) AB,(ii) (AB)T(iii) BTAT;(b) Pre-multiply each side of the equation BX = c by A.
Given the three matricesverify the associative law and the distributive law over addition.
Show that the transformationwith θ = 60° maps the square with cornersonto a square.
In quantum mechanics the components of the spin of an electron can be repesented by the Pauli matricesShow that
A rectangular site is to be levelled, and the amount of earth that needs to be removed must be determined. A survey of the site at a regular mesh of points 10 m apart is made. The heights in metres above the level required are given in the following table.It is known that the approximate volume of
A contractor makes two products P1 and P2. The four components required to make the products are subcontracted out and each of the components is made up from three ingredients A, B and C as follows:The cost per unit of the ingredients A, B and C are a, b and c respectively. The contractor makes the
(a) Given the matrixverify thatand show that 2A2 = A + I.(b) By repeated application of this result show also that for any integer n An = αA + βI for some a, b.
Find the values of x that make the matrix Z5 a diagonal matrix, where
Evaluate the third-order determinant
Evaluate the minors and cofactors of the determinantassociated with the first row, and hence evaluate the determinant.
Evaluate the 3 × 3 determinants
Given the matricesevaluate(a) |A|,(b) |B|(c) |AB|.
Evaluate
Illustrate the use of cofactors in the expansion of determinants on the matrix
Derive the adjoint of the 2 × 2 matricesand verify the results in (5.13), (5.14) and (5.15).Data from (5.13), (5.14) and (5.15).
Givendetermine adj A and show that A(adjA) = (adj A)A =|A| I.
Find A–1 and B–1 for the matrices
Givenevaluate (AB)–1, A–1B–1, B–1A–1 and show that (AB)–1 = B–1A–1
Given the two matricesshow that the matrix T–1AT is diagonal.
Write the five sets of equations in matrix form and decide whether they have or do not have a solution.
Find a solution of
Find the values of k for which the equationshave a non-trivial solution.
Find the values of λ and the corresponding column vector X such that (A – λI)X = 0 has a non-trivial solution, given
A function u(x, y) is known to take values u1, u2 and u3 at the points (x1, y1), (x2, y2) and (x3, y3) respectively. Find the linear interpolating function u = a + bx + cy within the triangle having its vertices at these three points.
Solve the matrix equation AX = c where
Use elementary row operations and elimination to solve the set of linear equations
Use the tridiagonal procedure to solve
Using elimination and back substitution, solve the equations
Solve
Solve the matrix equationby Gaussian elimination with partial pivoting.
Solve, by elimination, the equations
Find the rank of the matrices
Reduce the following equations to echelon form, calculate the rank of the matrices and find the solutions of the equations (if they exist):
Determine whether the following sets of vectors are linearly dependent or independent
Find the characteristic equation and the eigenvalues of the matrix
Find the characteristic equation for the matrix

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