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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
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
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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