A question in finding invertion

[transgalactic]

i am given an operator
S:R3[x]->R3[x]

and we have the polinomial from which we take the eigenvalues from
t^3 - 4t
find wether S invertable or not?i tried to think about that and i got that the aigenvalues are 2 , 0, -2
but that only prooves that it diagonizable

i know a law that if the determinant differs zero then its invertable
but i don't know how to apply it here
?

 

[Dick]

If it has a zero eigenvalue then it has a nontrivial kernel. It's not invertible.

 

[HallsofIvy]

i know a law that if the determinant differs zero then its invertable

Hopefully, you know that works both ways because it is the other way you need here: If the determinant of a matrix is 0, then it is not invertible.

When you diagonalize a matrix, its determinant stays the same. The determinant of a diagonal matrix is the product of the numbers on its diagonal. Since the eigenvalues are 2, 0, and -2, the "diagonalized" matrix would have those numbers on its diagonal and so its determinant is 0.

Of course, Dick's method works just as well.

 

[transgalactic]

ahhh because its determinant is the multiplication of the diagonal
and we got a zero in there
then the whole thing is zero

thanks