Let $V$ be the vector space of all $2$ by $2$ matrices. Let $<M_1, M_2>$ $=$ $tr(M_1^TM_2)$ be an inner product defined on $V$. Let $A$ $=$ $\begin{pmatrix}1&1\\ 1&0\end{pmatrix}$ be one of the matrices defined in this inner product. Find a basis for the orthogonal complement of this inner product.
So I let the other matrix simply be $B$ $=$ $\begin{pmatrix}a&c\\ b&d\end{pmatrix}$. I then did the following multiplication:
$\begin{pmatrix}1&1\\ 1&0\end{pmatrix}\begin{pmatrix}a&c\\ b&d\end{pmatrix}=\begin{pmatrix}a+b&c+d\\ \:a&c\end{pmatrix}$
So, to be an orthogonal complement, I would need for the trace of this new matrix to be $0$. This would mean that $a+b+c$ $=$ $0$. But I was stuck on how to simplify this down further as there are no other variables with known values which I can compare these three unknowns to. I understand that I need to find a more simplified form of matrix $B$ but I am struggling to figure out how to do that.
Moreover, I do see that the answer is $\left\{\begin{pmatrix}-1&-1\\ 0&0\end{pmatrix},\:\begin{pmatrix}0&-1\\ 1&0\end{pmatrix},\:\begin{pmatrix}0&0\\ 0&1\end{pmatrix}\right\}$, but I am not sure how this came about, especially since the trace of only the middle matrix in the basis is zero.
Any help or guidance would be highly appreciated!