Amazing matrices

$A$ is similar to some diagonal $D$ , i.e.:

$\large \displaystyle A = P^{-1}DP$

For some invertible matrix $P$ . So:

$\large \displaystyle A^n = P^{-1}DP \cdot P^{-1}DP \cdot \ ... \ \cdot P^{-1}DP \ \ \text{(n times) }$

$\large \displaystyle A^n = P^{-1}D^nP$

But also:

$\large \displaystyle A^n = 0$

Thus:

$\large \displaystyle P^{-1}D^nP = 0$

Multiplying by $P^{-1}$ to the right and by $P$ to the left on both sides:

$\large \displaystyle D^n = 0$

Since $D$ is diagonal:

$\color{#20A900} \boxed{\large \displaystyle D = 0}$

By the definition of $A$ :

$\large \displaystyle A = P^{-1}DP$

$\large \displaystyle A = P^{-1}0P$

$\color{#3D99F6} \boxed{\large \displaystyle A = 0}$