Multiplication-Free Gaussian Elimination and Matrix Inversion via Bitplane Semantics: Exact Trailing Updates with Boolean GEMM, GF(2) Gauss–Jordan, Bareiss, and Modular CRT

We extend our multiplication-free bit-sliced paradigm from matrix multiplication and determinants to Gaussian elimination and matrix inversion. All trailing updates are bilinear and can be executed by a Boolean (bit-sliced) GEMM with bitwise AND, population count, shifts and additions, yielding zero scalar multiplications at the matrix level. We integrate the bit-sliced core with three exact pivot/inversion regimes: (i) fully Boolean Gauss-Jordan over F2; (ii) fraction-free Bareiss over Z; (iii) modular LU over primes with CRT reconstruction. We provide executable code and small numerical checks; all GEMM-shaped updates are multiplication-free.