Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 36
![35. (a) Show that the eigenvalues of the 2 X 2 matrix
- [a b]
c d
A =
are the solutions of the quadratic equation
A² tr(A)λ + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
λ = ² ( a + d ± √(a − d)² + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
tr(A) = ₁ + A₂ and det A = d₁d₂
where A₁ and A₂ are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F46695490-535e-4e6d-be60-a5ac8c464218%2Ffb303ff6-f590-4944-bdfb-a3353310cd61%2F1i55o9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:35. (a) Show that the eigenvalues of the 2 X 2 matrix
- [a b]
c d
A =
are the solutions of the quadratic equation
A² tr(A)λ + det A = 0, where tr(A) is the trace
of A. (See page 162.)
(b) Show that the eigenvalues of the matrix A in
part (a) are
λ = ² ( a + d ± √(a − d)² + 4bc)
(c) Show that the trace and determinant of the matrix A
in part (a) are given by
tr(A) = ₁ + A₂ and det A = d₁d₂
where A₁ and A₂ are the eigenvalues of A.
36. Consider again the matrix A in Exercise 35. Give
conditions on a, b, c, and d such that A has
(a) two distinct real eigenvalues,
(b) one real eigenvalue, and
(c) no real eigenvalues.
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