<p>Bicomplex numbers, with their commutative structure and idempotent representation, offer a tractable alternative to quaternions in hypercomplex analysis and have attracted interest in recent years. This work fills a gap about bicomplex matrix factorizations like LU, QR, Cholesky and the SVD factorizations. We explore the Cauchy–Schwarz inequality which uses the partial order in hyperbolic numbers. By exploiting the idempotent representation, we also look at the pseudoinverse of bicomplex matrices and Sylvester’s matrix equation. We simplify analysis and highlight emerging applications, opening pathways for further research in bicomplex linear algebra.</p>