I wrote out A as a matrix looking like this:
$$\begin{bmatrix}a_1&a_2&a_3\\a_4&a_5&a_6\end{bmatrix}$$
And then using the equation $A\times[1\;1\;1]^T=[0\;0]^T$ I came to this system of equations:
$$\begin{cases}
a_1+a_2+a_3=0\\
a_4+a_5+a_6=0
\end{cases}
$$
So basically $A$ could be a matrix like this:
$$\begin{bmatrix}0&1&-1\\-3&7&-4\end{bmatrix}$$
or this:
$$\begin{bmatrix}-10&6&4\\-10&-5&15\end{bmatrix}$$
etc…
So I am able to find examples of such $A's$ but I don't know how to find all these $A's$.
I also need to find the dimension and basis for the linear space that these matrices create – the dimension is $6$ but what is the basis?
Best Answer
Notice that:
$$\begin{cases} a_1+a_2+a_3=0\\ a_4+a_5+a_6=0 \end{cases}$$
is a system of $2$ equations and $6$ variables. We can introduce $6-2 = 4$ real parameters, $x$, $y$, $z$ and $w$, such that:
$$\begin{cases} a_1 = -x-y\\ a_2 = x\\ a_3 = y\\ a_4 = -z-w\\ a_5 = z\\ a_6= w \end{cases}.$$
Therefore, all matrices $A$ which satisfy you requirements are in the form:
$$\begin{bmatrix}-x-y&x&y\\-z-w&z&w\end{bmatrix},$$
for given parameters $x, y, z$ and $w$.