I'm trying to teach myself linear algebra from this book. I'm on the eigenvalue/eigenvector chapter, and there's a problem in there that I've been stuck on, It defined the companion matrices (C) for 2nd, 3rd and 4th order matrices. It says that det(C-rI) is equal to the characteristic polynomial and I see that this is true for 2nd and 3rd order matrices. But then I have problems with the 4th order ones. So I got that C - rI would be -r 1 0 0 0 -r 1 0 0 0 -r 1 c₀ c₁ c₂ (c₃ - r) So according to my book a 4th order characteristic polynomial is r⁴ - c₃r³ - c₂r² - c₁r¹ - c₀ When I take det(C - rI) I get (-r)(-r)(-r)(c₃ - r) + (1)(1)(1)(c₀) + (0)(0)(0)(c₁) + (0)(0)(0)(c₂) - (c₀)(0)(1)(0) - (c₁)(-r)(0)(-r) - (c₂)(1)(0)(1) - (c₃ - r)(0)(-r)(0) This all simplifies to r⁴ - c₃r³ + c₀ Which isn't the characteristic polynomial So finally my question, Where did I go wrong, and please don't think I came here without first trying to get it on my own. So if you know this stuff could you please help me out. -Andrew [edited by - atcdevil on July 1, 2002 9:14:23 AM]

I''m a bit lazy (plus my time is short right now), so I don''t want to do the math on this, but my suspicion is that you''ve incorrectly written the formula for the determinant of the 4x4 matrix C-rI.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.