Show that if A is invertible and diagonalizable,then A^−1 is

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In summary, if A is invertible and diagonalizable, then A^-1 is diagonalizable. To find a 2×2 matrix that is not a diagonal matrix, not invertible, but diagonalizable, D must have a zero on the diagonal, making A not invertible. Therefore, D must be invertible to have a diagonalizable A^-1.
  • #1
charlies1902
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Show that if A is invertible and diagonalizable,
then A^−1 is diagonalizable. Find a 2 ×2 matrix
that is not a diagonal matrix, is not invertible, but
is diagonalizable.


Alright, I am having some trouble with the first part.
So far, I have this:
If A is diagnolizable then
A=PDP^-1 where P is the matrix who's columns are eigenvectors and D is the diagonal matrix of eigevenvalues of A.


(A)^-1=(PDP^-1)^-1
A^-1=PDP^-1

How's that?
 
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  • #2


charlies1902 said:
Show that if A is invertible and diagonalizable,
then A^−1 is diagonalizable. Find a 2 ×2 matrix
that is not a diagonal matrix, is not invertible, but
is diagonalizable.Alright, I am having some trouble with the first part.
So far, I have this:
If A is diagnolizable then
A=PDP^-1 where P is the matrix who's columns are eigenvectors and D is the diagonal matrix of eigevenvalues of A.(A)^-1=(PDP^-1)^-1
A^-1=PDP^-1

How's that?

Just because D is diagonal doesn't mean D=D^(-1). And after you've fixed that, how do you know D is invertible?
 
Last edited:
  • #3


Oh oops.
So
A^-1=P * D^-1 * P^-1

hmm, does D have to be invertible?
Can't you have eigen values of 0 and 2
so D looks like this:
0 0
0 2
which is not invertible?
 
  • #4


charlies1902 said:
Oh oops.
So
A^-1=P * D^-1 * P^-1

hmm, does D have to be invertible?
Can't you have eigen values of 0 and 2
so D looks like this:
0 0
0 2
which is not invertible?

They told you A is invertible. Doesn't that mean D has to be invertible? Can you prove that?
 
  • #5


Dick said:
They told you A is invertible. Doesn't that mean D has to be invertible? Can you prove that?

I'm confused on why you would have to prove that D is invertible and if it is always invertible.
I'm calling D the diagonal matrix who's diagonal elements are the eigenvalues of A.
 
  • #6


charlies1902 said:
I'm confused on why you would have to prove that D is invertible and if it is always invertible.
I'm calling D the diagonal matrix who's diagonal elements are the eigenvalues of A.

D isn't invertible if it has a zero on the diagonal. But if it does then A has a zero eigenvalue and it's not invertible. I'm not sure whether you have to prove that or whether you can just say it. But it's not hard to prove.
 

FAQ: Show that if A is invertible and diagonalizable,then A^−1 is

What does it mean for a matrix to be invertible and diagonalizable?

A matrix is invertible if it has an inverse matrix, which is a matrix that, when multiplied with the original matrix, results in the identity matrix. A matrix is diagonalizable if it can be written as a diagonal matrix by finding a basis of eigenvectors.

How do you show that if A is invertible and diagonalizable, then A-1 is also invertible and diagonalizable?

To show that A-1 is invertible, we can use the fact that the inverse of a diagonal matrix is also a diagonal matrix. To show that A-1 is diagonalizable, we can use the fact that the eigenvectors of A are also eigenvectors of A-1.

Can a matrix be invertible but not diagonalizable?

Yes, a matrix can be invertible but not diagonalizable. For example, the identity matrix is invertible but cannot be diagonalized as it is already a diagonal matrix.

Is it possible for a matrix to be diagonalizable but not invertible?

Yes, it is possible for a matrix to be diagonalizable but not invertible. For example, a zero matrix is diagonalizable as it is already in diagonal form, but it does not have an inverse.

How can the invertibility and diagonalizability of a matrix be useful in practical applications?

The invertibility and diagonalizability of a matrix can be useful in various applications, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and performing transformations in computer graphics. Invertible and diagonalizable matrices also have special properties that make them easier to work with and can provide insights into the behavior of a system or data set.

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