Showing that a certain binary matrix cannot be congruent to the null matrix

Question

I don't get why the following matrix (whose entries belong to the binary field) \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

cannot be congruent to the null matrix ,according to my notes.

Can I show this by resorting to the fact that the determinant of the product is the product of determinant? Is this result even true for binary matrices? If not, how can I show this?

Answer 1

If $A$ is congruent to $B$ then $A=P^{T}B P$ for some invertible matrix $P$

Suppose that $A=\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$ is congruent to the null matrix then by definition $$A=P^{T} 0 P$$ where $0$ is the null matrix and $T$is the traspose operator.

Hence form our assumption $$A=0$$ which is a contradiction by our given $A$ (or also can use the determinant both sides and conclude that $det A=det 0=0$ which is false since $detA\neq 0$).

Therefore $A$ can´t be congruent to the null matrix.