Comment by kens

Determinant as volume of the transformation is cool, but I still don't "get" the determinant intuitively despite many years of math. In particular, why does the determinant work to solve linear equations (i.e. Cramer's rule)? And what's the motivation behind the formula for the determinant? (I realize these questions are a bit vague, but I'm hoping for a more intuitive answer than "that's just the way the math works out".)