1. a) elimination b) [" rank. c) L 4 Solve the following system of linear equations using Gaussian method: 2x1 4x2 + 2x3 = -32 2x28x3 = -8 -4x1 +5x2 +9x3 = 79 Consider the matrix of coefficients of the above system and find its Are the rows of the coefficients matrix linearly independent?
1. a) elimination b) [" rank. c) L 4 Solve the following system of linear equations using Gaussian method: 2x1 4x2 + 2x3 = -32 2x28x3 = -8 -4x1 +5x2 +9x3 = 79 Consider the matrix of coefficients of the above system and find its Are the rows of the coefficients matrix linearly independent?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve this questions
![Show all your work in detail
1. a)
A Solve the following system of linear equations using Gaussian
elimination method:
b) ["
rank.
c) L
2.
2x1
4x2 + 2x3 = -32
2x2
823=-8
-4x1 +5x2 +9x3 = 79
Consider the matrix of coefficients of the above system and find its
Are the rows of the coefficients matrix linearly independent?
...] Find the inverse of the following matrix using the cofactor formula
3
1
-2 1
10
A =
3. Consider the following matrix
d) [
e)
of A. Show that AX = XD.
120
30-1
24 2
-10 3
s] Find the characteristic equation of A.
Find the eigenvalues of A.
Find the corresponding eigenvectors of A.
Why is A diagonalizable?
Let X be the matrix of eigenvectors and D be the diagonal form
A =
AX = XX
(A-XI) x = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54518529-781a-4abd-8f95-1d3f6e1337cd%2F1a09611c-740d-4c90-8730-85440ed5ef08%2Ffsvalqk.jpeg&w=3840&q=75)
Transcribed Image Text:Show all your work in detail
1. a)
A Solve the following system of linear equations using Gaussian
elimination method:
b) ["
rank.
c) L
2.
2x1
4x2 + 2x3 = -32
2x2
823=-8
-4x1 +5x2 +9x3 = 79
Consider the matrix of coefficients of the above system and find its
Are the rows of the coefficients matrix linearly independent?
...] Find the inverse of the following matrix using the cofactor formula
3
1
-2 1
10
A =
3. Consider the following matrix
d) [
e)
of A. Show that AX = XD.
120
30-1
24 2
-10 3
s] Find the characteristic equation of A.
Find the eigenvalues of A.
Find the corresponding eigenvectors of A.
Why is A diagonalizable?
Let X be the matrix of eigenvectors and D be the diagonal form
A =
AX = XX
(A-XI) x = 0
![4. a) [
Curl(F), and then div(Curl(F)).
Consider the vector field F
b)
For a given scalar field f= f(x, y, z) which is continuously differ-
entiable, prove that Curl(grad(f))==0.
5. T
Let F= [-y³,2³], R be the region defined by x² + y² ≤8 with
x,y 20 and C be the boundary of R in the counterclockwise orientation. Using
Green's theorem, evaluate the line integral fF.dr.
4+2√+Z÷1
١٠٤٣ و
Compute first
6.
Let S be the portion of the plane x+2y+z=2 in the first octant.
Evaluate the surface integral of the vector field F= [x2, xy, y²] over S.
Let F= [r sin² (y), y + z, z cos² (y)] and S be the surface defined
by x² + y² + z² = 9. Using the divergence theorem of Gauss, evaluate the surface
integral
U+2v
Js
F-n dA.
E-2-4-2N
Multiple choice questions
Choose the correct answer
for each of the following questions. Write your answer in the booklet.
MCQ 1. A point P(a, -a, 1) is located over a rotating body and its velocity
vector is v= [1,1,0]. If the angular velocity is w = [0,0, 1], then a is equal to
A. -1 B. -2 C. 1 D. 2 E. None of these
v=wxY
MCQ 2. If the rank of an n x n matrix A is equal to n-1, then
A. One row of A is zero B. The matrix A is not invertible C. det(A) #0
D. Rows of A are linearly independent E. None of these
MCQ 3. A curve is given by the parametrization r(t) = [-t2, t, t³] and the
tangent vector at some point A is [-2,1,3]. Then the point A is given by
342
A. (-1,1,1) B. (1,-1,-2) C. (-1,1,2) D. (-2,1,3) E. None of these
MCQ 4. If the vector field v= [mz, -my, z] is irrotational, then m is equal to
curl=0
A. 1 B. -1 C0 D. 1 E. None of these
u
MCQ 5. Let A and B be n x n matrices with det(A) = 4 and det (B) = 3. If
det (AT B*)= 4/27, then k is equal to
AETAK
AT
A.-2 B. 2 C. 3 D-3 E. None of these](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54518529-781a-4abd-8f95-1d3f6e1337cd%2F1a09611c-740d-4c90-8730-85440ed5ef08%2Fry7dwj4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. a) [
Curl(F), and then div(Curl(F)).
Consider the vector field F
b)
For a given scalar field f= f(x, y, z) which is continuously differ-
entiable, prove that Curl(grad(f))==0.
5. T
Let F= [-y³,2³], R be the region defined by x² + y² ≤8 with
x,y 20 and C be the boundary of R in the counterclockwise orientation. Using
Green's theorem, evaluate the line integral fF.dr.
4+2√+Z÷1
١٠٤٣ و
Compute first
6.
Let S be the portion of the plane x+2y+z=2 in the first octant.
Evaluate the surface integral of the vector field F= [x2, xy, y²] over S.
Let F= [r sin² (y), y + z, z cos² (y)] and S be the surface defined
by x² + y² + z² = 9. Using the divergence theorem of Gauss, evaluate the surface
integral
U+2v
Js
F-n dA.
E-2-4-2N
Multiple choice questions
Choose the correct answer
for each of the following questions. Write your answer in the booklet.
MCQ 1. A point P(a, -a, 1) is located over a rotating body and its velocity
vector is v= [1,1,0]. If the angular velocity is w = [0,0, 1], then a is equal to
A. -1 B. -2 C. 1 D. 2 E. None of these
v=wxY
MCQ 2. If the rank of an n x n matrix A is equal to n-1, then
A. One row of A is zero B. The matrix A is not invertible C. det(A) #0
D. Rows of A are linearly independent E. None of these
MCQ 3. A curve is given by the parametrization r(t) = [-t2, t, t³] and the
tangent vector at some point A is [-2,1,3]. Then the point A is given by
342
A. (-1,1,1) B. (1,-1,-2) C. (-1,1,2) D. (-2,1,3) E. None of these
MCQ 4. If the vector field v= [mz, -my, z] is irrotational, then m is equal to
curl=0
A. 1 B. -1 C0 D. 1 E. None of these
u
MCQ 5. Let A and B be n x n matrices with det(A) = 4 and det (B) = 3. If
det (AT B*)= 4/27, then k is equal to
AETAK
AT
A.-2 B. 2 C. 3 D-3 E. None of these
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