Cayley-Hamilton T

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.6: Matrices
Problem 6E
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The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if A is a 3 x 3 matrix with the
characteristic equation
Co+ Cz1 + C2A² + A3 = 0
then col + c,A + C2A² + A³ = 0, so
A3 = - c2A? - C1A - col
Multiplying through by A yields A4 = - CoA - CA² – C2A³, which expresses A4 in terms of A3, A? and A. Use this procedure to calculate
A3 and A4 for
0.
A =
0 1
1 -4 4
A³ is
Aª is
Transcribed Image Text:The Cayley-Hamilton Theorem provides a method for calculating powers of a matrix. For example, if A is a 3 x 3 matrix with the characteristic equation Co+ Cz1 + C2A² + A3 = 0 then col + c,A + C2A² + A³ = 0, so A3 = - c2A? - C1A - col Multiplying through by A yields A4 = - CoA - CA² – C2A³, which expresses A4 in terms of A3, A? and A. Use this procedure to calculate A3 and A4 for 0. A = 0 1 1 -4 4 A³ is Aª is
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