Eigenvalues

Nov 2019
2
0
UK
Hello

I know the eigenvalues for my matrix A and A^(-1) (I assume it is just 1/eigenvalue of A, is this right?)

but it says what are the eigenvalues for 2(A(^-1)) + I,

can anybody help? I don't have a clue.

TIA
 
Oct 2018
105
78
USA
If $A$ is invertible with eigenvalue $\lambda$, then $\frac{1}{\lambda}$ is an eigenvalue of $A^{-1}$.

$\displaystyle Av = \lambda v$
$\displaystyle v = \lambda A^{-1}v$
$\displaystyle \frac{1}{\lambda} v = A^{-1}v$

For this specific problem, just multiply by $v$ and go from there.

$\displaystyle (2A^{-1}+I)v = 2A^{-1}v + v = \frac{2}{\lambda}v + v = \left(\frac{2}{\lambda}+1 \right)v$

So the eigenvalue for $2A^{-1}+I$ would be $\frac{2}{\lambda}+1$.