# Determinant

• Dec 10th 2009, 06:12 PM
treetheta
Determinant
Let A be a 3x3 matrix which satifies the equation:

A^3 -3A^2 - 10A = 0

a) Find all the real numbers which could possibly be eigenvalues of A

alright... so what i did was simplify the equation and got (A)(A-5)(A+2)=0

but I don't know where to go from there

and

b) Assume, in addition that detA = 20. Find det(A^2 -3A)

I know what to do if i had the matrix but i dont so im kinda lost on this one

thanks
• Dec 10th 2009, 06:19 PM
mr fantastic
Quote:

Originally Posted by treetheta
Let A be a 3x3 matrix which satifies the equation:

A^3 -3A^2 - 10A = 0

a) Find all the real numbers which could possibly be eigenvalues of A

alright... so what i did was simplify the equation and got (A)(A-5)(A+2)=0

but I don't know where to go from there

and

b) Assume, in addition that detA = 20. Find det(A^2 -3A)

I know what to do if i had the matrix but i dont so im kinda lost on this one

thanks

a) Therefore $\displaystyle \lambda = 0, 5, -2$ since a matrix satisfies its characteristic equation (Cayley-Hamilton theorem).

b) Left multiply $\displaystyle A^3 -3A^2 - 10A = 0$ by $\displaystyle A^{-1}$ and simplify: $\displaystyle A^2 - 3A = 10 I$. Therefore ....
• Dec 10th 2009, 06:24 PM
treetheta
the determinant is equal to 1000

10(10-0) = 100

wow, your so smart =D but how would i ever think of that thanks, for the tip though it will help me look out for tricks