The Determinant through LU: Shears Keep It, Pivots Give It
An interactive live film: elimination shears (determinant one) carry the box of A's columns to U without changing its size, so det A is the product of the pivots. Drag a column until the box is flat.
Narration
Here is a three by three matrix A. Its three columns span a slanted box, and its determinant is the signed size of that box.
Gaussian elimination subtracts a multiple of one row from another. That is multiplying by a lower triangular matrix with ones on its diagonal, a shear, whose determinant is one. Watch the box slide into shape: its size never changes.
After elimination, A has become the upper triangular U. The determinant of a triangular matrix is the product of its diagonal, so the determinant of A is the product of the pivots.
One more rule: swapping two rows flips the sign of the determinant.
If the third column falls into the plane of the other two, the box goes flat, the last pivot is zero, and so is the determinant.
Your turn. Drag the tip of the green column across its level plane. Flatten the box: make the determinant zero.
Elimination keeps the determinant, and the pivots hand it to you: multiply the diagonal of U.
