A question in finding invertion

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In summary, the conversation discusses finding whether a given operator is invertible or not based on its eigenvalues. It is determined that if the determinant of the matrix is 0, then it is not invertible. The conversation also mentions using diagonalization to find the determinant, and how having a zero eigenvalue leads to a zero determinant.
  • #1
transgalactic
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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
?
 
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  • #2
If it has a zero eigenvalue then it has a nontrivial kernel. It's not invertible.
 
  • #3
transgalactic said:
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.
 
  • #4
ahhh because its determinant is the multiplication of the diagonal
and we got a zero in there
then the whole thing is zero

thanks
 

Related to A question in finding invertion

What is inversion in science?

Inversion in science refers to the process of reversing or flipping the order of a sequence of elements. In genetics, it can refer to the reversal of the normal order of genes on a chromosome.

Why is finding inversion important?

Finding inversion is important because it can provide valuable information about the structure and function of genetic material. It can also help scientists understand the causes of genetic disorders and identify potential treatments.

What techniques are used to find inversion?

There are several techniques used to find inversion, including karyotyping, fluorescent in situ hybridization (FISH), and genomic sequencing. These techniques involve analyzing the arrangement and sequence of genetic material to identify any inversions.

What are the potential impacts of inversion on an organism?

Inversions can have a variety of impacts on an organism, depending on the specific genes and sequences involved. Some inversions may have no noticeable effect, while others can lead to genetic disorders or changes in physical traits.

Can inversion be inherited?

Yes, inversion can be inherited from one generation to the next. However, the specific inheritance pattern may vary depending on the type and location of the inversion. In some cases, inversions may be passed down through multiple generations without causing any noticeable effects.

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