[Shri R Ganeshram]

James,
My message was the mathematical fact. :) I think the inverse matrix talk inspired that ("primed me").
It was not the solution to the second part.
"The determinant of an invertible matrix is the same as the determinant of that matrix after undergoing elementary row operations. Aka, given matrix A, invertible, and a set of row operations R, and that B is R(A), det A= det B.

I'm not sure how well I phrased that, but I think the point of the exercise was to draw from knowledge and not from an outside source. :)

-Shri"
-Shri