Alternative method to the matrix problem?

[rock.freak667]

Homework Statement

Find all real matrices,X, of the form

[x 0]
[y z]

such that X²+3X+2I=0

(I is the identity matrix)

Homework Equations

The Attempt at a Solution

The only method I know is to work out X² and then 3X, add to 2I, then equate the corresponding entries with the zero matrix. Have 3 equations with three unknowns and solve. But is there any other method in which I can use to shorten the amount of writing I will have to do?

 

[morphism]

If X^2+3X+2I=0, then the minimal polynomial of X divides t^2+3t+2=(t+1)(t+2), and take it from here.

 

[rock.freak667]

If X^2+3X+2I=0, then the minimal polynomial of X divides t^2+3t+2=(t+1)(t+2), and take it from here.

I was thinking to do something like that, but the identity matrix in it confused me.
Now I should re-write the equation as

X^2++X+2I^2=(X+I)(X+2I)=0 and go from here right?

 

[morphism]

Well, it depends on what you mean by "go from here"! If you're going to say something like "Then X=-I or X=-2I", then you'll be off the mark. Do you know what a minimal polynomial is?

 

[rock.freak667]

Well, it depends on what you mean by "go from here"! If you're going to say something like "Then X=-I or X=-2I", then you'll be off the mark. Do you know what a minimal polynomial is?

Yes, that is what I thought to do... No idea what a minimal polynomial is.

 

[morphism]

OK, do you know what eigenvalues are, then?

By the way, for matrices, AB=0 doesn't imply that A=0 or B=0. Try to come up with an example of this - and make sure you understand why this happens. In fact, you can have a nonzero matrix A such that A^2=0.

 

[rock.freak667]

OK, do you know what eigenvalues are, then?

By the way, for matrices, AB=0 doesn't imply that A=0 or B=0. Try to come up with an example of this - and make sure you understand why this happens. In fact, you can have a nonzero matrix A such that A^2=0.

Yes,I know what eigenvalues are, and the characteristic polynomial. I am aware the AB=0 doesn't always mean that A=0 or b=0...I always forget this in matrices.

 

[morphism]

Then I guess a shortcut would be to notice that the eigenvalues of a triangular matrix will lie on its diagonal. And because X^2+3X+2I=0, the eigenvalues of X will satisfy the equation t^2+3t+2=0. So this'll give us why x and z will be. To find y you'll just have to do it the long way.

Well come to think of it, this isn't really a shortcut at all.

 

[rock.freak667]

I guess I will have to stick with my initial method.
My math course, doesn't include the topic of e.vectors/values but since I did Further math I just happen to know it.