How do you solve this matrix to diagonalize it?

Question

How do you guys solve this? I tried the characteristic polynomial. Found an eigenvalue of 0, and a,b,c=0. Am I using the correct approach? Are there other ways to solve this?

Answer 1

You found that the only eigenvalue of $A$ is zero. The matrix $A$ will be diagonalizable if and only if you can find four linearly independent eigenvectors for the only possible eigenvalue zero. In other words, $A$ will be diagonalizable if and only if the kernel of $A$ will be of dimension four. That is, if and only if $A = 0$ (or $a = b = c = 0)$.

Answer 2

Hint: The matrix $A$ has a single eigenvalue $0$ with algebraic multiplicity $4$; in order to be diagonalizable, the geometric multiplicity must be $4$ as well, so the rank of $A=A-0I_4$ must be…