3.14 Consider the companion-form matrixA =-α11-α2010-α3001-α4000 ela Show that its characteristic polynomial is given by2.15A(2)=24+a₁2³ +₂2² +α32+α4PROBLEMS 97Show also that if λ; is an eigenvalue of A or a solution of A(2) = 0, then [^^ λ; 1]' isan eigenvector of A associated with λ₁.

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Answered: 3.14 Consider the companion-form matrix…

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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5 Find the specified change-of-coordinates matrix. 7) Let B and C be bases for R2, where Find the change-of-coordinates matrix from B to C. b1 = ¹₁ -[3]- ¹₂ -[-+-C₁ -[3]· <₂2-[-₁6] b2 = C1 = C2 =
8. Transform the augmented matrix M to triangular matrix. -1 -1 4 -1 (A) 0 0.67 4.33 3.33 (B) 0 0.67 -4.33 3.33 Riz Ri-(RX 21) 0 6.50 12.0 -1 -1 0₁-6.50 12.0 -1 -1 (C) 0 -0.67 4.33 (D) 0 -0.67 -4.33-3.33] (3-4-14 છે. 0 lo 0 -6.50 -12.01 201² + 2001 2x142 1 300309 -3.33 6.50 -12.0 Els M=4 -2 3 2 L2 1 -5 -1
-1 Show that A = 1 invertible matrix P satisfying A = P-¹BP. P-1 = 3 - and B = -33 -1 P= are similar matrices by finding an
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2 1 2.1. Given the matrix of 0 = 0 1 Prove that (OP)-¹ = P-10-¹. 1 3 0 and matrix of P = 2 -1 -1 2 5 3 1 1 1
2. A nilpotent matrix is a square matrix A € Mnxn (F) such that Ak = 0, for some integer k. -3 1 0 2 3 -1 (b) Prove that every nilpotent matrix is not invertible. (a) Show that the matrix B = 1 2 - is nilpotent.
4) Determine whether the given set of functions is linearly independent on the interval (-x,). a) (z) = cos 2z, (2) = 1, 1,()= cos z b) (z)=z, 1,(2)=z-1, (z)=z+3 c) ()=e**, 1,(=)=e
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6) If X = 3 is a matrix of order 2x1, given that A= 1 find the matrix X such B = that AX =3B+2C. [1 5 marks 1
3
4. (i) Consider the 3 x 3 permutation matrix 0 1 0 P = |0 0 1 0 0 Find all 3 x 3 matrices A such that PAPT = A. %3D (ii) Consider the 4 × 4 permutation matrix 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0, P = Find all 4 x 4 matrices A such that PAPT = A.

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3.14 Consider the companion-form matrix A = -α1 1 0 0 -α2 0 1 0 -α3 0 0 1 -α4 0 0 0