Differential Equations: An Introduction to Modern Methods and Applications
3rd Edition
ISBN: 9781118531778
Author: James R. Brannan, William E. Boyce
Publisher: WILEY
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Textbook Question
Chapter A.4, Problem 19P
In each of Problems 17 through 20, use a computer to find the eigenvalues and eigenvectors for the given matrix.
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Problem 7
Determine the eigenvalues and eigenspaces of the matrix
A :=
120
0 1 0
-1 2-2
Problem Eight
Diagonalize this matrix, whose eigenvalues are 2 and 8. Note: You only need to find P and D.
Adjust the eigenvectors (by scaling) so that they have a 1 in the lowest nonzero position.
[4 2 21
A = 2 4 2
2 2 4
Solution:
a. Determine the solutions of the simultaneous equations given below
y = x² + x + 1
y = 6x + 7
b. Solve the following inequality:
|3x + 1| <3
c. Solve the following equation using logarithm, correct to 3 significant figures.
33t-1 = 7t+1
3
d. Simplify the matrix multiplication
2 -1
-4
7 31
e. Find the inverse matrix of Matrix A =
=[-_-2-3]
6
5
f. Find the determinant of the Matrix B= 2
-2
-4
NÁG
5
2
-2
ܢܐ ܗ ܚ
6
-3
1
0
3 -3
X
م بل ه
-3
0
Chapter A Solutions
Differential Equations: An Introduction to Modern Methods and Applications
Ch. A.1 - Given the matrices...Ch. A.1 - If A=(120321213) and if B=(102011213), find...Ch. A.1 - Demonstrate that A=(223101111) and B=(112011102)...Ch. A.1 - Prove each of the following laws of matrix...Ch. A.1 - 5. If , under what conditions is to be...Ch. A.1 - 6. Prove that sums and products of upper(lower)...Ch. A.1 - Let A=diag(a11,.....ann) be a diagonal matrix....Ch. A.1 - Prove that if A is symmetric and nonsingular, then...Ch. A.1 - Two square matrices A and B are said to commute if...Ch. A.1 - 10. If is any square matrix, show each of the...
Ch. A.2 - In each case, reduce A to row reduce echelon form...Ch. A.2 - In each of Problems 2 through 5, if there exist...Ch. A.2 - In each of Problems 2 through 5, if there exist...Ch. A.2 - In each of Problems 2 through 5, if there exist...Ch. A.2 - In each of Problems 2 through 5, if there exist...Ch. A.2 - In each of Problems 6 through 9. Find the general...Ch. A.2 - In each of Problems 6 through 9. Find the general...Ch. A.2 - In each of Problems 6 through 9. Find the general...Ch. A.2 - In each of Problems 6 through 9. Find the general...Ch. A.2 - In each of Problems 10 through 14, determine...Ch. A.2 - In each of Problems 10 through 14, determine...Ch. A.2 - In each of Problems 10 through 14, determine...Ch. A.2 - In each of Problems 10 through 14, determine...Ch. A.2 - In each of Problems 10 through 14, determine...Ch. A.2 - In each of Problems 15 through 17, determine...Ch. A.2 - In each of Problems 15 through 17, determine...Ch. A.2 - In each of Problems 15 through 17, determine...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - In each of Problems 1 through 10, use elementary...Ch. A.3 - Let and
Verify that .
Ch. A.3 - If A is nonsingular, show that |A1|=1/|A|.Ch. A.3 - In each of Problems 13 through 15, find all values...Ch. A.3 - In each of Problems 13 through 15, find all values...Ch. A.3 - In each of Problems 13 through 15, find all values...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each of Problems 1 through 10, find all...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each Problems 11 through 16, find the...Ch. A.4 - In each of Problems 17 through 20, use a computer...Ch. A.4 - In each of Problems 17 through 20, use a computer...Ch. A.4 - In each of Problems 17 through 20, use a computer...Ch. A.4 - In each of Problems 17 through 20, use a computer...
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- Problem #7: Find the eigenvalues of the following matrix. A = 6 2 -2 1 6 2arrow_forwardFor Problems 1-5, use Equation (7.1.1) to verify that A and v are an eigenvalue/eigenvector pair for the given matrix A. 5. 1 = 10, v= c1(1, –4, 0) + c2(0, 0, 1), 6 -1 A -16 6 0 where cj and c2 are con- -4 -1 10 stants.arrow_forwardFIND the eigenvalues of the following matrix: [−3 −1] B= [ 2 −5]arrow_forward
- If the eigenvalues of a 3×3 matrix ? is 1, −8 and 50, then the eigenvalues of ?T are 1, −8 and 50. True or false?arrow_forwardIn this problem, if you give decimal answers then give at least three digits of accuracy beyond the decimal. The matrix has the following complex eigenvalues (give your answer as a comma separated list of complex numbers; use "i" for ✓-1 and feel free to use a computer to solve the relevant quadratic equation): λ = 1.65+1.548386257i, 1.65-1.548386257i Since A has non-real eigenvalues, it is not diagonalizable, but we can find a matrix C = section 5.5 of Lay): C = P = cos(6) A = sin(8) -sin(8) cos(8) [113] 1.8 Further, we may factor Cas C = XY, where X is a matrix that scales by the positive real number radians, with —π/2 ≤0 ≤ π/2) is and an invertible matrix P such that A = PCP-¹ (I want you to cook up C, P as in and Y is rotation matrix whose counter-clockwise rotation angle (in This means that if we let B be the basis for R² consisting of the columns of P, then the B-matrix of A is C. So if we're willing to change our basis for IR², then the linear transformation x Ax really is just…arrow_forwardPlease help. For problem 4, find the characteristic polynomials and eigenvectors/eigenvalues. Thank you.arrow_forward
- You are provided with the following matrix B (where the 2nd and 3rd rows are not given), and vector x. Given that x is an eigenvector of the matrix A, what would be the corresponding eigenvalue? B = -10 -10 10 x= 2 * * * 4 * * * 4arrow_forwardAnswer the following. Show the solution. 2 -2 -4 A = |-2 1 4 51 1. Eigenvalues of matrix A. 2. Determinant of matrix A. 3. Inverse of matrix A. 4. Transpose of matrix Aarrow_forwardThe eigenvalues of the matrix A=[1 3 ] [4 2 ]. are 5 and −2 (you are not asked to prove this, you can use it without proof). For each real number x, find the rank of the matrix xI−A=[ x−1 -3 ] [ −4 x−2].arrow_forward
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