I'm currently preparing for an exam, and I'm getting stuck at this seemingly straightforward question. The question is to check if the matrix $A$, given below, is congruent and/or orthogonal equivalent to a diagonal matrix.
$$A = \begin{bmatrix} -3 & -2 & 0 \\ -1 & -1 & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & 2 & 0 \\ 1 & -3 & 0 \\ 0 & 0 & 1\end{bmatrix}$$
The issue I'm having is that all theorems in the text that I'm using are aimed toward symmetric/skew-symmetric/hermitian/skew-hermitian matrix when it comes to congruent matrices.
Since this matrix $A$ has none of those properties, the only way I know to check congruence to a diagonal matrix is by using elementary row operations. Yet given the way $A$ is presented, I feel like I'm missing a straightforward solution. Is there an easier way to show the matrix is congruent and/or orthogonal equivalent to a diagonal matrix?