Every unit matrix is a LULU

The four matrices L0U0L1U1 at the end of the title are triangular with ones on their main diagonals. Their product has determinant one. Following a question and theorem of Toffoli, we show that any matrix with determinant one can be factored in this way. A transformation of the plane becomes a sequence of one-dimensional shears, with n2 — 1 free parameters.