Values of a,b,c that make a matrix diagonalizable

Question

For which, if any, values of $a,b,c \in \mathbb{R}$ is the matrix

diagonalizable?

The Eigenvalues are easy enough. I get $\lambda=1$ (With an algebraic multiplicity of $2$) and $\lambda=c$ (With an algebraic multiplicity of $1$). I'm not really sure how to proceed though.

I could get the eigenvectors by imposing restrictions on the eigenvalues but that's a very painful process and I am not sure if this is the best way too approach this... Any suggestions?

Answer 1

Find the minimal polynomial of $M$. $M$ will be diagonalisable if and only if the minimal polynomial is a product of linear factors.