1. ²²(²5 3) ` = [(-3³ -¯2)] a[; Find x such that 2
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
![1.
2.
Find x such that
1
2x
¹2²[ ²5 ] ^ - [ -3³ -2²]
=
3
Solve the following systems of three equations in three
variables by determining the inverse of the matrix of
coefficients and then using matrix multiplication.
3. Write down the matrix corresponding to the given transformation R² → R².
(a) Reflection across the line y = -x and then rotation by 7/3 radians counter clockwise.
(b) Projection onto the line y = x/4 and then dilation by a factor 2.
6.
7.
(a) x₁ + 2x₂ - x₂ = 2
x₁ +
x₂ + 2x₂ = 0
X₂ - x₂ = 1
3 1 4
4. Compute the determinant of the matrix-7-2 1
9 1 -1]
[4 9 14]
5. I performed elementary row operations on a matrix A and obtained the matrix 0-3 -7 The
Lo 0 2
operations I performed were (1) swapping two rows, (2) multiplying all rows by two, and (3) adding five
times one row to another row. What was the determinant of the original matrix.
In Exercises 15 and 16 determine the characteristic polynomi-
als, eigenvalues, and corresponding eigenspaces of the given
4 X 4 matrices.
15.
4
1
0
2-2 2
3 1 -1
0 2 0
1-3 5
16.
35-5 5
3 1 3-3
-2 2 0 2
0 4-6 8
Prove that the constant term of the characteristic poly
mial of a matrix A is (A.
8.
Let A be a matrix with eigenvalue À having corresponding
eigenvector x. Let c be a scalar. Prove that A - c is an eigen-
value of A- c/ with corresponding eigenvector x.
9. Prove that the set of all functions f: [0,1] → R that satisfy ff(x) dx = 0 is a vector space under the
operations of pointwise addition and scalar multiplication.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5cf73431-844e-4c23-9b17-47dbe521c237%2F8b101529-d527-4c40-aa18-5eaa4d7fd50c%2Fe30xx2k_processed.png&w=3840&q=75)
Transcribed Image Text:1.
2.
Find x such that
1
2x
¹2²[ ²5 ] ^ - [ -3³ -2²]
=
3
Solve the following systems of three equations in three
variables by determining the inverse of the matrix of
coefficients and then using matrix multiplication.
3. Write down the matrix corresponding to the given transformation R² → R².
(a) Reflection across the line y = -x and then rotation by 7/3 radians counter clockwise.
(b) Projection onto the line y = x/4 and then dilation by a factor 2.
6.
7.
(a) x₁ + 2x₂ - x₂ = 2
x₁ +
x₂ + 2x₂ = 0
X₂ - x₂ = 1
3 1 4
4. Compute the determinant of the matrix-7-2 1
9 1 -1]
[4 9 14]
5. I performed elementary row operations on a matrix A and obtained the matrix 0-3 -7 The
Lo 0 2
operations I performed were (1) swapping two rows, (2) multiplying all rows by two, and (3) adding five
times one row to another row. What was the determinant of the original matrix.
In Exercises 15 and 16 determine the characteristic polynomi-
als, eigenvalues, and corresponding eigenspaces of the given
4 X 4 matrices.
15.
4
1
0
2-2 2
3 1 -1
0 2 0
1-3 5
16.
35-5 5
3 1 3-3
-2 2 0 2
0 4-6 8
Prove that the constant term of the characteristic poly
mial of a matrix A is (A.
8.
Let A be a matrix with eigenvalue À having corresponding
eigenvector x. Let c be a scalar. Prove that A - c is an eigen-
value of A- c/ with corresponding eigenvector x.
9. Prove that the set of all functions f: [0,1] → R that satisfy ff(x) dx = 0 is a vector space under the
operations of pointwise addition and scalar multiplication.
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