1. ## [SOLVED] Eigenvalues

Hi again

Just after an answer check if that's OK

The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix $\displaystyle (2A-3I)^-1$

I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me $\displaystyle \frac{-1}{5}, -1, 1$

Is this correct?

Thanks

2. Hello
Originally Posted by Ian1779
Hi again

Just after an answer check if that's OK

The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix $\displaystyle (2A-3I)^-1$

I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me $\displaystyle \frac{-1}{5}, -1, 1$

Is this correct?

Thanks
Yes it is correct !!!

3. Originally Posted by Ian1779
Hi again

Just after an answer check if that's OK

The eigenvalues of Matrix A are -1,1 and 2. What are the eigenvalues of the Matrix $\displaystyle (2A-3I)^-1$

I have got the eigenvalues of 3I to be 3,3,3 so in the equation and then the inverse gives me $\displaystyle \frac{-1}{5}, -1, 1$

Is this correct?

Thanks
I can't say anything about your shortcut, but after working out the matrices I get your result.

-Dan

4. Originally Posted by topsquark
I can't say anything about your shortcut, but after working out the matrices I get your result.

-Dan
The identity matrix is always diagonal in any basis.

If a matrix A has these eigenvalues, then in a certain basis, the matrix A will be $\displaystyle \begin{pmatrix}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}$

Consider we're on this basis, we have :

$\displaystyle 2A-3I=\begin{pmatrix} -5&0&0 \\ 0&-1&0 \\ 0&0&1 \end{pmatrix}$

And the inverse of a diagonal matrix is the diagonal matrix formed by the inverses of the original diagonal.