Is this a binary operation? For the following set, determine if the given operation is a binary operation or not. The set of all 2 × 2 matrices with real entries whose 2, 2 entry is zero: $$X=\left\\{\begin{bmatrix}a & b \\\c & 0\end{bmatrix} | a,b,c \in \mathbb{R}\right\\}$$ With matrix multiplication. Normally id say yes right away but the part thats confusing me is that after its multiplied by anther matrix its very likely that the 0 entry will no longer be a 0 entry does that disqualify it?

> after its multiplied by anther matrix its very likely that the 0 entry will no longer be a 0 entry does that disqualify it?

Yes. A binary operation on a set $X$ is a function $X \times X \rightarrow X$. So the output of multiplying _any_ two such matrices must also be a matrix with a zero in the $2,2$ slot. You can investigate this by multiplying together two general matrices in this set:

$$ \left[ \begin{matrix} a & b \\\ c & 0 \end{matrix} \right] \cdot \left[ \begin{matrix} d & e \\\ f & 0 \end{matrix} \right] = \left[ \begin{matrix} ad + bf & ae \\\ cd & ce \end{matrix} \right]$$

Given that $c$ and $e$ are arbitrary...