How is the matrix $A^{2}$ related to $a\times(a\times b)$?

Question

I have a matrix, $$A = \left( \begin{matrix} 0&-a_3& a_2\\ a_3&0&-a_1\\ -a_2&a_1&0\end{matrix}\right)$$

So, $$A^{2} = \left( \begin{matrix} -a_3^{2}-a_2^{2}&a_2a_1& a_3a_1\\ a_1a_2&-a_3^{2}-a_1^{2}&a_3a_2\\ a_1a_3&a_2a_3&-a_2^{2}-a_1^{2}\end{matrix}\right)$$

I know that $a=\left( \begin{matrix}3\\2\\1\end{matrix}\right)$ and $b=\left( \begin{matrix}1\\2\\3\end{matrix}\right)$

Subbing in $a_1=3$, $a_2=2$ and $a_3=1$ into $A^2$, I get

$$A^{2} = \left( \begin{matrix} -5&6&3\\ 6&-10&2\\ 3&2&-13\end{matrix}\right)$$

I've also determined that $a\times(a\times b) = \left( \begin{matrix}0\\16\\-32\end{matrix}\right)$

I'm told $a\times(a\times b)$ can be found using $A^{2}$, but I don't see the connection. What am I missing?

Answer 1

Try to verify by direct computation that given any $x \in \mathbb{R^3}$, $$Ax=a\times x$$