eigenvalues of a matrix $A$ plus $cI$ for some constant $c$

If $A$ is a $n \times n$ real matrix with eigenvalues $\lambda_1,\lambda_2,...\lambda_n$, how does one get the eigenvalues of the matrix $A$ + c$I$, where $I$ is the identity matrix and $c$ is a non-zero real constant?

I tried to work out the characteristic polynomials, but I am wondering if there is a way to quickly get the eigenvalues.

Solution 1:

I'd order the proof correctly: $Ax=\lambda x$ and $cIx = cIx$, so $Ax + cIx =\lambda x +cIx$. Then rearrange to get $(A+cI)x=(\lambda+c)x$.

Solution 2:

Note that if $Av = \lambda v$, then $(A+cI)v = (\lambda+c)v$.

Also, $\chi_{A+cI}(x) = \det (xI-A-cI) = \det( (x-c)I -A) = \chi_A(x-c)$.