The most important value the determinant can take is zero. A determinant of zero means the transformation squashes the whole plane down onto a single line (or even a point) — it destroys a dimension. Once that happens, the move can never be undone: countless different starting vectors get crushed onto the same output, so there's no way to tell where any of them came from.

A matrix you can reverse is called invertible (or non-singular). The clean test is simply:

Watch a dimension die

Drag the slider to swing the second column until it lines up with the first. As they become parallel, the determinant slides to zero and the entire grid flattens onto one line. At that instant the matrix is singular: no inverse exists.

One number, many meanings

"\det A = 0" is one of the busiest sentences in mathematics. It says all at once: the columns are linearly dependent; the transformation isn't reversible; the linear system A\vec{x}=\vec{b} has either no solution or infinitely many; and the columns fail to span the plane. Whenever any one of those is true, they are all true — and the determinant is the single dial that detects it.