For which integer values of n is the matrix diagonalizable by a real matrix?

Question

I need to find the values of n for which matrix C is diagonalizable by a real matrix.

$$ C=\begin{pmatrix} 1 & 1\\ n & n + 1\\ \end{pmatrix} $$ I've calculated the characteristic polynomial $λ^2 - 2λ - λn + 1 = 0$

Which turns to $λ^2 - λ(2 + n) + 1 = 0$

At this step I'm lost. What do I do afterwards?

Answer 1

Hint The discriminant is $(n+2)^2-4=n(n+4)$

in $(-4,0)$ it is negative the matrix is not a real diagonalizabe.

and if $n<4, n>0$ it is positive, there exists two distinct eigenvalues the matrix is diagonalizabe.

Study the case $n=0,n=4$ when it has a double root