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 = =
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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![[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
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