Nondiagonal $3 \times 3$ matrix

Question

Can someone give an example of a nondiagonal, $3 \times 3$ matrix that is diagonalizable but is not invertible?

Explanation would be appreciated.

Answer 1

Start off by getting a diagonal matrix $A$ with a $0$ on the diagonal so it isn't invertible.

For instance consider $A=\begin{bmatrix} 0 & 0 &0\\ 0 &1 &0\\ 0 & 0 & 1 \end{bmatrix}$.

Now you want a non diagonal matrix. Just consider $B^{-1}AB$ for some appropiate $B$. Some $B$ not too simple.

A possibility is $B=\begin{bmatrix} 1 & 2 &3\\ 0 &4 &5\\ 0 & 0 & 6 \end{bmatrix}$. It clearly is invertible and $A\sim B^{-1}AB=\begin{bmatrix} 0 & -2 &-3\\ 0 &1 &0\\ 0 & 0 & 1 \end{bmatrix}$.

Therefore a possible answer is $B^{-1}AB$.

Now find your own $A$ and $B$.

Answer 2

$A = \begin{bmatrix}−3&−2&0\\2&−2&0\\13&-4&0\end{bmatrix}$

For the structure of the matrix, you easily can derive an approach for any dimension.

See it?