19. As in Exercise 18, calculate A¹0x for 1 2-1 05-2 6-2 A = 0 and x = 2 4]. 7

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.7: The Inverse Of A Matrix
Problem 30E
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Linear algebra: please solve q19 correctly and handwritten

18. If a vector x is a linear combination of eigenvec-
tors of a matrix A, then it is easy to calculate the
product y = Akx for any positive integer k. For in-
stance, suppose that Au₁ = λ₁u₁ and Au₂ = λ₂u₂,
where u₁ and u₂ are nonzero vectors. If x =
a₁u₁ + a₂u₂, then (see Theorem 11 of Section 4.4)
y = Akx = Ak (a₁µ₁ +a₂µ₂) = a₁ Aku₁ +9₂ Aku₂
α₁ (λ₁)*µ₁ + a₂(λ₂)ku₂. Find A¹0x, where
A =
4-2
A =
5-3
]
and x =
[;]
19. As in Exercise 18, calculate A¹0x for
1 2 -1
05-2
06-2
and x =
2
4
7
Transcribed Image Text:18. If a vector x is a linear combination of eigenvec- tors of a matrix A, then it is easy to calculate the product y = Akx for any positive integer k. For in- stance, suppose that Au₁ = λ₁u₁ and Au₂ = λ₂u₂, where u₁ and u₂ are nonzero vectors. If x = a₁u₁ + a₂u₂, then (see Theorem 11 of Section 4.4) y = Akx = Ak (a₁µ₁ +a₂µ₂) = a₁ Aku₁ +9₂ Aku₂ α₁ (λ₁)*µ₁ + a₂(λ₂)ku₂. Find A¹0x, where A = 4-2 A = 5-3 ] and x = [;] 19. As in Exercise 18, calculate A¹0x for 1 2 -1 05-2 06-2 and x = 2 4 7
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