Proving a Matrix's Invertibility with Eigenvalue 0: Understanding Determinants

[transgalactic]

prove that a matrix cannot be invertible if it has eigen value of 0.

not invertible mean that the determinant equals 0.

x(x-2)=0
??

 

[Mark44]

prove that a matrix cannot be invertible if it has eigen value of 0.

not invertible mean that the determinant equals 0.

x(x-2)=0
??

Where does x(x -2) come in?

You have an n x n matrix A that has 0 as an eigenvalue. How do you find the eigenvalues of a matrix? How are eigenvalues defined?

 

[transgalactic]

x(x -2) its just an example for a polynomial which is not inertible
i thought of proving like this

that if an operator T is not invertible then det T=0 then dim ker differs 0
(we have a row of zeros) so i have eigen value 0

correct
??

 

[transgalactic]

Where does x(x -2) come in?

You have an n x n matrix A that has 0 as an eigenvalue. How do you find the eigenvalues of a matrix? How are eigenvalues defined?

"If an equation containing a variable parameter possesses nontrivial solutions only for certain special values of the parameter, these solutions are called eigenfunctions and the special values are called eigenvalues."

how to use it
??

 

[Office_Shredder]

Not the kind of eigenvalue you want (well, sort of is, but let's not complicate things). An eigenvalue $$\lambda$$ of a matrix A is a number such that there exists v in the vector space, v non-zero, with

$$Av = \lambda v$$

 

[transgalactic]

so if i have eigen value zero
Av=0

how does it show that its not invertible
i know that now v plays the role of kernel A
but i don't know what is the link between the kernel of an operator and its
ability to be invertible
??

 

[Office_Shredder]

A matrix is invertible if it has... an inverse. To have an inverse, it must be 1-1. Think about what having a non-trivial kernel means with regards to this

 

[transgalactic]

1-1 means that for each x i have a y value and vise versa
i really don't know what is the affect of kernel
i know how to find it in a matrix
i know
dim I am + dim Ker =V

??

 

[statdad]

Remember that in order for a vector $$v$$ to be an eigenvector, it must be true that $$v \ne 0$$. If that is the case, yet [tex]A v = 0 $$, what can you conclude?$$[/tex]

 

[transgalactic]

zero vector is the product of multiplication of the operarator by the kernel vectors

i don't know
??

 

[statdad]

Well, remember that if the matrix $$A$$ is invertible, the solution to

$$Ax = b$$

is always given by

$$x = A^{-1}b$$

Now, suppose that $$v \ne 0$$ is an eigenvector that corresponds to the eigenvalue $$\lambda = 0$$

Now assume that $$A$$ is invertible. What does the equation

$$Av = \lamba \, v = 0$$

tell you about $$v$$? What does this contradict, and what assumption led to that contradiction?

 

[Dr.D]

Can you use the fact that the inverse is the matrix of cofactors divided by the determinant, and that division would be undefined if you had a zero eigenvalue because the determinant is the product of the eigenvalues?

 

[maze]

Consider
Ax = 0
A0 = 0

What is A^-1(0)?

 

[transgalactic]

A matrix is invertible if it has... an inverse. To have an inverse, it must be 1-1. Think about what having a non-trivial kernel means with regards to this

what is the answer
??

 

[transgalactic]

why if dim ker differs 0
then its not invertible
??

 

[Mark44]

Let's get organized. You're trying to prove this statement, which is a paraphrase of your first post in this thread:

If matrix A has 0 as an eigenvalue, A does not have an inverse.​

If A has 0 as an eigenvalue, Ax = 0x = 0, for some nonzero vector x (the eigenvector associated with eigenvalue 0).

So Ax = 0, where x is not the zero vector.
From this equation, is it possible to say either that A definitely has an inverse or that A definitely doesn't have an inverse?

 

[transgalactic]

i can say that definitely we have a kernel vector in A
but i don't know how it affects the invertiblity of A
??

 

[Mark44]

There is always at least one vector in ker(A), regardless of whether A has an inverse.

 

[transgalactic]

yes 0,0,0,

 

[HallsofIvy]

This has already been said, but I'll try again. If Ax= 0 and A has an inverse, what do you get if you multiply both sides of that equation by A^-1? What does that tell you about the kernel of A? How is that a contradiction?

 

[transgalactic]

i get x=0
what does it mean?

 

[HallsofIvy]

That means that "x= 0" is the only solution to Ax= 0. What does that tell you about the kernel of A?