The great moment of our first matrix factorization is here. In this piece, we work through a 3x3 example showing how row operations—elimination and swaps—solve $\mathbf{A}\vec{x}=\vec{b}$ by turning the problem into two triangular systems. Triangular systems are easy to solve. Lower triangular systems are unpacked with forward subsititution (new), upper ones with back substitituion (old). Including row swaps with a single permutation matrix $\mathbf{P}$, our factorization is $\mathbf{P}\mathbf{A}=\mathbf{L}\mathbf{U}$ for any matrix $\mathbf{A}$. Because Matrixology eschews any attempt at branding, this factorization is known as $\mathbf{L}\mathbf{U}$ factorization. Let's call it Triangular Factorization.