The Determinant

What to observe

1. The two arrows are the columns of A: where î and ĵ land. The unit square between the old ones is carried to the parallelogram between the new ones, and the determinant is nothing more exotic thanits area. Drag a tip and watch the number track the shading.
2. Turn on base times height. The area is the length of one arrow times how far the other stands off its line, and the dashed leg is that height. Slide the second arrow along the direction of the first: the height never changes, so neither does the area. That single fact is why adding a multiple of one row to another leaves a determinant alone.
3. Push the arrows towards each other until they line up. The parallelogram thins to nothing, det hits 0, and the whole plane has been pressed onto a line. Everything on that line now has infinitely many preimages and everything off it has none, which is precisely whatnon-invertible means. The determinant answers that question with a single number.
4. Keep pushing past the collapse. The shading changes colour and the determinant comes out negative. Turn on A shape inside the square: the little F flips into its mirror image. The sign is not an accident of the formula, it is the record of whether the plane was turned over, and to get from a right-handed frame to a left-handed one you have to pass through zero, which is what Animate shows on the Reflect preset.
5. The square is a convenient object, not a special one. With the shape on, watch the row for its area: whatever it started as, it comes out multiplied by the same |det A|. The determinant is a property of the map, not of the thing being mapped.
6. Now switch B on. Doing A and then B scales area twice over, sodet(BA) = det B · det A falls out with no algebra at all. Set B to Reflect while A is already a reflection: two flips undo each other, two minus signs multiply to a plus, and the F comes back the right way round.