For which values of $a$, $b$, and $c$ is the following matrix diagonalizable?
$$\begin{bmatrix} 1 & 0 & c \\ 1 & a & b \\ 0 & 0 & 1 \end{bmatrix}$$
As I understand it, in order for matrix to be diagonalizable, it be written in the form:
$$A = P \Lambda P^{-1}$$
where $\Lambda$ contains the eigenvalues and the columns of $P$ are the eigenvectors.
I have calculated the eigenvalues which are : $\lambda_1 = 1, \lambda_2 = a$, but I got stuck after that. Any help?
Case $1$: $a=1$.
Consider the matrix $A-I$,
$$A-I=\begin{bmatrix} 0 & 0 & c \\ 1 & 0 & b \\ 0 &0 &0\end{bmatrix}$$
The nullity of the matrix is less than $3$,the algebraic multiplicity corresponding to the eigenvalue $1$ , it cannot be diagonalizable.
Case $2$: $a \neq 1$:
Again, consider the matrix $$A-I=\begin{bmatrix} 0 & 0 & c \\ 1 & a-1 & b \\ 0 &0 &0\end{bmatrix}$$
How does $a,b,c$ affects the nullity of the matrices?
To be diagonalizable, it has to be has nullity of $2$ (the algebraic multiplicity of eigenvalue $1$), i.e. the matrix $A-I$ has to be of rank $1$.
Are you able to complete the rest?