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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Find the inverse by Gauss€“Jordan (or by (4*) if n = 2). Check by using (1). calon calon
Solve the linear system given explicitly or by its augmented matrix. Show details. 01 -2 -3 -6 15 1 1 -1 2. 3. 2.
Showing all intermediate results, calculate the following expression or give reasons why they are undefined:AAT, A2, BBT, B2 Let 4 -2 3 -3 A =-2 1 B = -3 6. 1 2 2 -2 1. a = [1 -2 0], 0), b = 3 2 -2
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.(C + D) + E, (D + E) + C, 0(C - E) + 4D, A - 0C Let 4 A 6. B = 3
Show the following:Rank BTAT = rank AB. (Note the order!)
Showing the details, evaluate: c| b.
Find the inverse transformation. Show the details.y1 = 3x1 + 2x2y2 = 4x1 + x2
Solve the linear system given explicitly or by its augmented matrix. Show details. Г2 3 1 -11 -2 -4 -1 3 -3 3 3 4 -7 -7
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.(5u + 5v) - 1/2 w, -20(u + v) + 2w, E - (u + v), 10(u + v) + w Let
Find the inverse transformation. Show the details.y1 = 0.2x1 - 0.1x2y2 = -0.2x2 + 0.1x3y3 = 0.1x1 +
Showing the details, calculate the following expressions or give reason why they are not defined, whenA-1, B-1 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Find the following expression, indicating which of the rules in (3) or (4) they illustrate, or give reasons why they are not defined.15v - 3w - 0u, -3w + 15v, D - u + 3C, 8.5w - 11.1u + 0.4v Let 4 A
Find the value of the determinant of the n x n matrix An with main diagonal entries all 0 and all others 1. Try to find a formula for this. Try to prove it by induction. Interpret A3 and A4 as
Find the rank by Theorem 3 (which is not very practical) and check by row reduction. Show details. 4 -6 4 10 -9- 10
Showing the details, calculate the following expressions or give reason why they are not defined, when(A + AT)(B - BT) 3 1 -3 4 1 2, B =-4 A = 4 -2 -3 2 V =-3
Formula (4) is occasionally needed in theory. To understand it, apply it and check the result by Gauss€“Jordan:In Prob. 6Data from Prob. 6Find the inverse by Gauss€“Jordan (or by
Determine the equilibrium solution (D1 = S1, D2 = S2) of the two-commodity market with linear model (D, S, P = demand, supply, price; index 1 = first commodity, index 2 = second commodity)D1 = 40 -
Showing the details, find all solutions or indicate that no solution exists.5x - 3y + z = 72x + 3y - z = 08x + 9y - 3z = 2
Are the following sets of vectors linearly independent? Show the details of your work. [0.4 -0.2 0.2], [O 0 0], [3.0 -0.6 1.5]
Find all vectors in R3 orthogonal to [2 0 1]. Do they form a vector space?
Solve by Cramer’s rule. Check by Gauss elimination and back substitution. Show details. 3x - 2y + z = 13-2x + y + 4z = 11 x + 4y -
Showing the details, find all solutions or indicate that no solution exists.-6x + 39y - 9z = -122x - 13y + 3z = 4
The idea is that elementary operations can be accomplished by matrix multiplication. If A is an m x n matrix on which we want to do an elementary operation, then there is a matrix E such that EA is
Find all 2 X 2 matrices A = [αjk] that commute with B = [bjk], where bjk = j + k.
In a production process, let N mean “no trouble” and T “trouble.” Let the transition probabilities from one day to the next be 0.8 for N → N, hence 0.2 for N → T, and 0.5 for T → N,
Beginning with the last of the vectors [3 0 1 2], [6 1 0 0], [12 1 2 4], [6
Showing the details, find all solutions or indicate that no solution exists. -8x + 2z = 1 6y + 4z = 312x + 2y
In a community of 100,000 adults, subscribers to a concert series tend to renew their subscription with probability 90% and persons presently not subscribing will subscribe for the next season with
Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, v2, · · · denote components.)All vectors in R3 with 3v2 + v3 = k
Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, v2, · · · denote components.)All vectors in Rn with the first n - 2
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. [2.0 0.4] 2.0] [0.4
Find the eigenvalues. Find the corresponding eigenvectors.
Verify this for A and A = P-1AP. If y is an eigenvector of P, show that x = Py are eigenvectors of A. Show the details of your work. 7 -5 P = 10 %3D 2 -7 -1
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. -b
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors. -1 + i
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors.
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. cos 0 -sin e sin 0 os e cos
Find the eigenvalues. Find the corresponding eigenvectors. 4
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. [5 13
Given A in a deformation y = Ax, find the principal directions and corresponding factors of extension or contraction. Show the details. 1.25 0.75 0.75 1.25
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. Га a
Is the given matrix Hermitian? Skew-Hermitian? Unitary? Find its eigenvalues and eigenvectors. 2 + 2i 2 - 2i 2 + 2i 2 - 2i
Find the limit state of the Markov process modeled by the given matrix. Show the details. [0.4 0.3 0.3 0.1 0.3 0.6 0.3 3 0.1 0.6
Find the eigenvalues. Find the corresponding eigenvectors. a -b a
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. cos e - sin e cos e sin e
Find the growth rate in the Leslie model (see Example 3) with the matrix as given. Show the details. 9.0 5.0 0.4 0.4
Find the eigenvalues. Find the corresponding eigenvectors. - sin cos 0 0 sin 0 cos e
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details. -1
Are the following matrices symmetric, skew-symmetric, or orthogonal? Find the spectrum of each, thereby illustrating Theorems 1 and 5. Show your work in detail. 7. colo tlo -lo
Is the matrix A Hermitian or skew-Hermitian? Find xÌ…TAx. Show the details. [2i -2 + 3i i 2 + 3i
Find the eigenvalues. Find the eigenvectors. -7 – 12 -12 7. 4.
Find the growth rate in the Leslie model (see Example 3) with the matrix as given. Show the details. 3.0 2.0 2.0 0.5 0.5 0.1
Find the eigenvalues. Find the corresponding eigenvectors. 3 5 3 4 6. 1
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details. 7.7 -4.3 1.3 9.3
(a) Prove that the product of two orthogonal matrices is orthogonal, and so is the inverse of an orthogonal matrix. What does this mean in terms of rotations?(b) Show that (6) is an orthogonal
Is the matrix A Hermitian or skew-Hermitian? Find xÌ…TAx. Show the details. 1 i 4 1 -i i 3 4
Find an eigenbasis (a basis of eigenvectors) and diagonalize. Show the details. -5 -6 -9 -8 12 A1 = -2 –12 - 12 - 12 16
Show that a consumption matrix as considered in Prob. 13 must have column sums 1 and always has the eigenvalue 1.Prob 13Suppose that three industries are interrelated so that their outputs are used
Find the eigenvalues. Find the corresponding eigenvectors. [2
Find the eigenvalues. Find the eigenvectors. 2 -1 1 -1 8.5 2.
What kind of conic section (or pair of straight lines) is given by the quadratic form? Transform it to principal axes. Express xT = [x1 x2] in terms of the new coordinate
Find div v and its value at P.v = [0, cos xyz, sin xyz], P: (2, 1/2π, 0]
What curves are represented by the following? Sketch them.[α + t, b + 3t, c - 5t]
Find the components of the vector v with initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector u in the direction of v.P: (1, 1, 1), Q: (2, 2, 0)
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = arctan (y/x)
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = x/(x2 + y2)
What happens in Example 5 if you choose a P at distance 2d from the axis of rotation?
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|a + c|2 + |a - c|2 - 2(|a|2 + |c|2)
Find div v and its value at P.v = (x2 + y2 + z2)-3/2[x, y, z]
What curves are represented by the following? Sketch them.[α + 3 cos πt, b - 2 sin πt, 0]
Find the terminal point Q of the vector v with components as given and initial point P. Find |v|.4, 0, 0; P: (0, 2, 13)
Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.f = (x2 - y2)/(x2 + y2)
Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work.v = [e-z2, e-x2, e-y2]
What are the velocity and speed in Prob. 7 at the point (4, 2, -2) if the wheel rotates about the line y = x, z = 0 with ω = 10 sec-1?Data from Prob. 7A wheel is rotating about the y-axis with
What curves are represented by the following? Sketch them.[cosh t, sinh t, 2]
Find the terminal point Q of the vector v with components as given and initial point P. Find |v|.13.1, 0.8, -2.0; P: (0, 0, 0)
Prove and illustrate by an example.∇̅(fg) = f∇g + g∇̅f
Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles).v = [sec x, csc x, 0]
What kind of surfaces are the level surfaces f(x, y, z) = const?f = 9(x2 + y2) + z2
Graph the given velocity field v of a fluid flow in a square centered at the origin with sides parallel to the coordinate axes. Recall that the divergence measures outflow minus inflow. By looking at
Find the terminal point Q of the vector v with components as given and initial point P. Find |v|.0, -3, 3; P: (0, 3, -3)
Prove and illustrate by an example.∇2(fg) = g∇2f + 2∇f • ∇g + f ∇2g
Let v be the velocity vector of a steady fluid flow. Is the flow irrotational? Incompressible? Find the streamlines (the paths of the particles).v = [-y, x, π]
What kind of surfaces are the level surfaces f(x, y, z) = const?f = z - √x2 + y2
What does u • v = u • w imply if u = 0? If u ≠ 0?
What does a × b = a × c with a ≠ 0 imply?
(a) What direction does curl v have if v is parallel to the yz-plane?(b) If, moreover, v is independent of x?
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.T = xy
Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.f = 9x2 + 4y2
Find curl v for v given with respect to right-handed Cartesian coordinates. Show the details of your work.v = [2y2, 5x, 0]
Verify Lagrange€™s identity for |a x b| for a = [3, 4, 2] and b = [1, 0, 2]. Prove it, using sin2Î³ = 1 - cos2 Î³. The identity is |a x b| = V(a • a) (b • b)
Let a = [1, -3, 5], b = [4, 0, 8], c = [-2, 9, 1].Find:|a + b|, |a| + |b|
Find div v and its value at P.v = [v1( y, z), v2(z, x), v3(x, y)], P: (3, 1, -1)]
What curves are represented by the following? Sketch them.[-2, 2 + 5 cos t, -1 + 5 sin t]
Find the components of the vector v with initial point P and terminal point Q. Find |v|. Sketch |v|. Find the unit vector u in the direction of v.P: (1, 4, 2), Q: (-1, -4, -2)
Find grad f. Graph some level curves f = const. Indicate ∇f by arrows at some points of these curves.(y + 6)2 + (x - 4)2
Model and solve by the Laplace transform:Find the model (the system of ODEs) in Prob. 38 extended by adding another mass m3 and another spring of modulus k4 in series.Data from Prob. 38Show that the
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 130, assuming zero initial current and charge and:R = 2 Î©, L = 1 H, C = 0.1 F, v = 255
Using the Laplace transform and showing the details, find the current i(t) in the circuit in Fig. 130, assuming zero initial current and charge and:R = 4 Î©, L = 1 H, C = 0.05 F, v = 34e-t
Model and solve by the Laplace transform:Show that the model of the mechanical system in Fig. 149 (no friction, no damping) is туй — -kiy1 + ka(y2 — Ул) mıyi т2у? — —k2(у — у) —

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