For which values of $a,b,c,d,e\in\mathbb{R}$, is this $4\times 4$ matrix diagonalizable?

Question

For which values of $a,b,c,d,e\in\mathbb{R}$, is the matrix $A$ diagonalizable?

Here,

$$A=\left( \begin{array}{cc} 3 & 0 & 0 & 0 \\ a & 2 & d & e \\ b & 0 & 1 & 0 \\ c & 0 & f & 0 \end{array} \right).$$

I found the eigenvalues:

$$\lambda_{1}=3,\lambda_{2}=f\,\textrm{(if $f\neq 0$)},\lambda_3=0,\lambda_{4}=e .$$

Do I really need to compute the eigenvectors to find (or not) some conditions about $a,b,c,d,e$? Or there is another way to approache this problem ?

Answer 1

You should obtain eigenvalues of 0,1,2,3. These are distinct and so the matrix is diagonalizable.