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If A is similar to B and A^3-5A+2I = 0, show that B^3-5B+2I = 0.?
Let A and B be square matrices of the same size. If A is similar to B and A^3-5A+2I = 0, show that B^3-5B+2I = 0.
I'm not exactly sure, Moon, but this was on my last quiz and it was worth 6 marks. I think I needed a lot more work done in the proof. Are there any steps I can use to show that B can replace A?
2 Answers
- MTLv 59 years agoFavorite Answer
A is similar to B => A = M⁻¹BM for some matrix M
A³ = (M⁻¹BM)(M⁻¹BM)(M⁻¹BM) = M⁻¹B³M
5A = 5M⁻¹BM
A³ - 5A + 2I = 0
<=> M⁻¹B³M - 5M⁻¹BM + 2I = 0
Left-multiply both sides by M
<=> MM⁻¹B³M - 5MM⁻¹BM + 2MI = 0
<=> B³M - 5BM + 2I = 0
Then right-multiply both side by M⁻¹
<=> B³MM⁻¹ - 5BMM⁻¹ + 2IM⁻¹ = 0
<=> B³ - 5B + 2I = 0
Q.E.D
<=>
- MoonLv 79 years ago
What more ..........?
Its has already been proved in the question.
A^3-5A+2I = 0,
A is similar to B
B can take place of A
as is already done in
B^3-5B+2I = 0.
not the slightest difference
B^3-5B+2I = 0. thus stands proved.
