Finding det(A) given A² - 3A - 2I = 0 and det(A - 3I) = 2

[bojo]

Homework Statement

Let A be a 3 x 3 matrix satisfying the equation $$A^{2}$$-3A-2I=0 where I is the 3x3 identity matrix. Find det(A) given the det(A-3)=2

The Attempt at a Solution

Well can't find anything like this in my textbook, notes or google. I imagine its a pretty simple matrix property I've overlooked but otherwise i have no clue what to do!


Help much appreciated,
Ben

 

[rock.freak667]

det(AB) =det(A)*det(B)

So if you factor A^2-3A you can use det(A-3) = 2

 

[bojo]

so

A^2 -3A = 2I factors to

A(A-3)=2I

and using property det(AB)=det(A)det(B)

det(A)det(A-3)=det(2I) so det(A)=det(2I)/det(A-3)

 

[rock.freak667]

Yes and you can easily find det(2I).