The lapped orthogonal transform (LOT) is a popular transform and has found many applications in signal processing. Its extension, the biorthogonal lapped transform (BOLT), has been investigated in detail by Vaidyanathan and Chen (see IEEE Trans. Signal Processing, p.1103-15, 1995). In this paper, we study the lapped unimodular transform (LUT). All of these three transforms are first-order matrices with FIR inverses. We show that like LOT and BOLT, all LUTs can be factorized into degree-one unimodular matrices. The factorization is both minimal and complete. We also show that all first-order systems with FIR inverses can be minimally factorized as a cascade of degree-one LOT, BOLT, and unimodular building blocks. However unlike LOT and BOLT, unimodular filter banks of any order (which include LUTs as a special case) can never have linear phase.