a b Let A = cd We have defined the determinant of A: det(A) = A = ad - bc. There is another quantity of interest for A: the trace of A is defined as: tr(A) = 7 = a + d. A) Use the characteristic polynomial and the quadratic equa- tion to show that the eigenvalues of A are λ = T± √7²-4A 2 Use the result of part (a) to explain how we can quickly decide if a 2 x 2 matrix has real or complex eigenvalues. Use your method from part (b) to quickly determine whether 2 3] has real or complex eigenvalues. -1 4 A = =
a b Let A = cd We have defined the determinant of A: det(A) = A = ad - bc. There is another quantity of interest for A: the trace of A is defined as: tr(A) = 7 = a + d. A) Use the characteristic polynomial and the quadratic equa- tion to show that the eigenvalues of A are λ = T± √7²-4A 2 Use the result of part (a) to explain how we can quickly decide if a 2 x 2 matrix has real or complex eigenvalues. Use your method from part (b) to quickly determine whether 2 3] has real or complex eigenvalues. -1 4 A = =
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 19E
Related questions
Question
100%
![[a b]
c d
}]
We have defined the determinant of A: det(A) = A =
ad bc. There is another quantity of interest for A: the trace of A is
defined as: tr(A) = T = a + d.
Let A =
A) Use the characteristic polynomial and the quadratic equa-
T± √T²-4A
tion to show that the eigenvalues of A are λ =
2
B) Use the result of part (a) to explain how we can quickly
decide if a 2 × 2 matrix has real or complex eigenvalues.
A =
-
2
-1
Use your method from part (b) to quickly determine whether
3
has real or complex eigenvalues.
4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba71b9f-6645-4b82-be5d-cda1c9d5ec57%2F6de074a7-3967-4a2c-b39b-e77f1a98e81a%2F2g9ex3b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[a b]
c d
}]
We have defined the determinant of A: det(A) = A =
ad bc. There is another quantity of interest for A: the trace of A is
defined as: tr(A) = T = a + d.
Let A =
A) Use the characteristic polynomial and the quadratic equa-
T± √T²-4A
tion to show that the eigenvalues of A are λ =
2
B) Use the result of part (a) to explain how we can quickly
decide if a 2 × 2 matrix has real or complex eigenvalues.
A =
-
2
-1
Use your method from part (b) to quickly determine whether
3
has real or complex eigenvalues.
4
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
