Operation Matrix

Let me examine you from every dimension
Before you start any identity extension,
And from your column, why what arose?
Is this really you or just a transpose?
It's elementary, my dear, operation reduce
To your purest form, no need to induce,
To invert, or expand - don't isolate me!
Though we associate, is this symmetry?
I won't lead you on, I'm not that type of one,
Some parameters are set - to come undone.
Not schewing results, consider, give it a try:
Your M by the N with my j by the i, let's multiply?
Prove it, it's clear, our determinant is defined;
It's okay to be singular, but matrix you mine.