Assume that A and B are in M(C). Show that if Bis invertible, then there exists a scalarCE C suchthat A + cB is not invertible.Hint: First, show that det (A + tB)- det(B) - det (AB+ tl). Next, use the fundamental theoremmust have a complex eigenvalue A. Conclude that A+ cBis-1of algebra to argue that the matrix ABnot invertible for c=A.

Answered: Image /qna-images/answer/4687f6b2-705f-44d2-9970-93b0d24ed5d3.jpg

Assume that A and B are in Ma(C). Show that if Bis invertible, then there exists a scalar e e C such that A+ cB is not invertible. Hint: First, show that det(A+ tB)- det (B) - det (AB + tla). Next, use the fundamental theorem of algebra to argue that the matrix AB not invertible for c must have a complex eigenvalue . Conclude that A+ cBis A.

Assume that A and B are in Ma(C). Show that if Bis invertible, then there exists a scalar e e C such that A+ cB is not invertible. Hint: First, show that det(A+ tB)- det (B) - det (AB + tla). Next, use the fundamental theorem of algebra to argue that the matrix AB not invertible for c must have a complex eigenvalue . Conclude that A+ cBis A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Let L: PP be defined by L(ar+br+c)3Dct+b. Find the matrix representing L with respect to the basis S=(r, 2-+1,2-r). (b) Find the eigenvalues and associated eigenvectors of L.
Suppose u1, ... , Un and v1, ... , Vn are orthonormal bases for Rn. Construct the matrixA=UI:VT that transforms each Vj intouj to giveAv1 =u1, ... , Avn =Un.
Suppose A has 0 as an eigenvalue. What can you say about the determinant of A?
Let A be a non-singular matrix, det(A) ± 0. a) Show that µ(A) does not change if the operator A is multiplied by an arbitrary constant. b) Multiply one row of the Matrix A by some scalar a, and denote the new matrix by Aq. Show that µ(Aª) → ∞ as a → ∞.
Consider the operator R2 → R2given by left-multiplication by the matrix: B =[6 −7, 1 −2]. Find the characteristic polynomial cB(x), the eigenvalues of B, and theeigenspaces of B.
Write down a n × n matrix A such that Cond(A) =1, but determinant(A) → 0 as n → ∞.
Prove that if the determinant of a square matrix "A" does not equal to zero, then zero can not be an eigen value of "A".
Please show all steps clearly, thank you.
I need the answer as soon as possible
(3) Let matrix A have eigenvectors . . associated with eigenvalues Xo = 4, A₁ = 2, A₂ = 0), respectively. Suppose some vector 7 is: 7 = №n +27₁ +30¹₂ Compute Ar. Give your result as a linear combination of eigenvectors ₁.
Write down anxn matrix A such that Cond(A) =1, but determinant(A) → 0 as n → ∞.
1)Be (1 1 -1 A = 2 0 1 1 1 0 a)Calculate the eigenvalues of A. b)Find the minimal polynomial of A and say whether 4 is diagonalizable.