Find $a,b,c $ so the matrix is diagonalizable

Question

I am supposed to find the values of $a,b,c \in R$ for which the matrix can be diagonalized.

$\begin {bmatrix} 1 && 0 && c \\ 1 && a && b \\ 0 && 0 && 1 \end{bmatrix}$

I know it can be diagonalized if it has 3 linear independent eigenvectors, but that job seems too complicated in this case. I am not sure how to approach this problem, so any help is appreciated. Thanks

Answer 1

In order to diagonalize a matrix, you want to be able to utilize row operations such that you only end up with values along the diagonal. An easy case is $a = 1, c, b = 0$. Try plugging in and using row operations to simplify the matrix to see why and see if you can generalize this.

Answer 2

Hint:

Starting point:

$$\begin {vmatrix} 1-\lambda && 0 && c \\ 1 && a-\lambda && b \\ 0 && 0 && 1-\lambda \end{vmatrix} = (a-\lambda) \begin {vmatrix} 1-\lambda && c \\ 0 && 1-\lambda \end{vmatrix}=(a-\lambda)(1-\lambda)^2$$

You can consider two cases

Case 1. $a =1$ and

Case 2. $a \neq 1$.

Let $\lambda=1$ and see how many eigenvectors corresponds to it.

That is study the nullity of the matrix

$$\begin {bmatrix} 1-\lambda && 0 && c \\ 1 && a-\lambda && b \\ 0 && 0 && 1-\lambda \end{bmatrix}$$