# Thread: Matrix characteristic equation help

1. ## Matrix characteristic equation help

let M=[3,1,0;1,2,1;0,1,3] where ; specifies a new row

show that the characteristic equation is f(x)=(3-x)(x-4)(x-1)

Find det(M-xI) which is:
det[(3-x),1,0;1,(2-x),1;0,1,(3-x)]
=(3-x)((2-x)(3-x)-1)-1((3-x)(0)
=(3-x)(3-x)(2-x)-(3-x)
This expands to -x^3+8x^2-20x+15..which is not the same as the expanded form of the given equation..which is -x^3+8x^2-19x+12....

where have i gone wrong...

Also would the eigenvalues for the characteristic equation be 3, 4 and 1?

2. Hello,

Originally Posted by deragon999
let M=[3,1,0;1,2,1;0,1,3] where ; specifies a new row

show that the characteristic equation is f(x)=(3-x)(x-4)(x-1)

Find det(M-xI) which is:
det[(3-x),1,0;1,(2-x),1;0,1,(3-x)]
=(3-x)((2-x)(3-x)-1)-1((3-x)(0)
Here is the mistake :

It's ok for $\displaystyle (3-x)((2-x)(3-x)-1)$
But the second term is $\displaystyle -1((3-x){\color{red}-}0)$

--> \displaystyle \begin{aligned} f(x) &=(3-x)((2-x)(3-x)-1)-1(3-x) \\ &=(3-x)(6-5x+x^2-2) \\ &=(3-x)(4-5x+x^2) \\ &=(3-x)(4-x)(1-x) \end{aligned}

Also would the eigenvalues for the characteristic equation be 3, 4 and 1?
Yes

3. ## Tks

Omg that makes me look like a dumbass after getting it right in the first half...
neways thanks for ur help on both my latest questions..and many others besides.