Show that if $A^3=0$ but $A^2\ne0$, then $A^2v=0$ has a nontrivial solution

Question

Question. Let $A$ be a $3\times 3$ matrix. If $A^3=0$ and $A^2 \neq 0$, prove that $A^2v=0$ for some $v\in\mathbb{R}^3\setminus0$.

To generalize the solution I'm defining $A$ as: \begin{pmatrix} a_1 & a_2 & a_3\\ a_4 & a_5 & a_6\\ a_7 & a_8 & a_9 \end{pmatrix}

and have calculated $A^2$ to be:

\begin{pmatrix} a_1^2+a_2a_4+a_3a_7 & a_1 a_2+a_2 a_5+a_3 a_8 & a_1 a_3+a_2 a_6+a_3 a_9\\ a_4 a_1+a_5 a_4+a_6 a_7 & a_4 a_2+a_5^2+a_6 a_8 & a_4 a_3+a_5 a_6+a_6 a_9\\ a_7 a_1+a_8 a_4+a_9 a_7 & a_7 a_2+a_8 a_5+a_9 a_8 & a_7 a_3+a_8 a_6+a_9^2 \end{pmatrix}

Just the thought of having to calculate $A^3$ gives me a headache... I've read something about diagnolization, but I couldn't apply it here.

I assume there must be some characteristic of a matrix $A$ for which $A^3=0$ I couldn't think about.

Answer 1

$$\det A^3 = 0 \Rightarrow \det A = 0 \Rightarrow \det A^2 = 0$$

Because $A^2$ is not invertible it has to have nontrivial kernel. Therefore there is $v$ that $A^2 v = 0$

You don't really need to know that $A^2 \neq 0$.

Answer 2

The question is about the rank of the matrix. The assumption $A^3=0$ tells you that $A^3$ has rank zero, in particular its determinant is zero. Now, by Binet's theorem, $\det A^3= \det A \det A^2=(\det A)^3$, and the first term being zero tessl you that both $A$ and $A^2$ have not maximal rank (null determinant). Now, having a null determinant is a necessari and sufficient condition for the existence of a nonzero $v$ such that $A^2 v=0$ (notice that if $v$ can be zero you can always choose one).

Answer 3

$A^{3}=0$ hence for any $v\in\mathbb{R}^{3}$ $A^{3}v=0$ thus $A^{2}(Av)=0$.

That is: Any vector of the form $Av$ is a solution $x$ for $A^{2}x=0$.

Since $A\neq0$ then there is some $x\in\mathbb{R}^{3}$ s.t $Ax\neq0$. This gives a non trivial solution to $A^{2}x=0$ (hence also shows it is not of full rank)

Answer 4

$$A^2 \begin{bmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$$

Therefore $$A^2 \begin{bmatrix} a_1 \\ a_4 \\ a_7 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \end{bmatrix}$$

At least one of the columns of $A$ isn't zero if $A^2$ isn't zero.