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Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix below. Find A8:
- a)225A - 254J
- b)-255A -254 J
- c)15A - 14J
- d)255A 254J
Correct answer is option 'A'. Can you explain this answer?
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Cayley-Hamilton Theorem states that a square matrix satisfies its own ...
Characteristic equation, - λ (3-λ) 2 = 0
λ2 - 3λ 2 = 0
So, A2 - 3A 2I = 0 .....(1)
⇒ New, A4 = A2 x A2
From equation (1),
A4 = (3A – 2I) × (3A – 2I)
= 9A2 – 12A 4I
= 9{3A – 2I} – 12A 4I
= 27A – 18I – 12A 4I
A4 = 15 A – 14I
Now, A8 = A4 × A4
= (15A – 14I) × (15A – 14I)
= 225 A2 – 420 A 196I
= 225 {3A – 2I} – 420 A 196I
A8 = 255 A – 254 I
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Cayley-Hamilton Theorem states that a square matrix satisfies its own characteristic equation. Consider a matrix below. Find A8:a)225A - 254Jb)-255A -254 Jc)15A - 14Jd)255A 254JCorrect answer is option 'A'. Can you explain this answer?
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