A = LU: the triangle U and the receipt L.
Elimination turns A into an upper triangular U. The multipliers you used fit into a lower triangular L, exactly where you used them. Multiply back and you get A.
Think: meal prep
On Sunday you chop every vegetable for the week. That is the slow part. On weekdays each dinner takes five minutes.
Factoring A = LU is the Sunday work. Each new right-hand side b is a weekday dinner: two quick triangular solves.
Why the multipliers drop right in
Elimination is E₃₂E₃₁E₂₁A = U. Undo the cards in reverse order and you get A = E₂₁⁻¹E₃₁⁻¹E₃₂⁻¹U. That product of inverses is L.
Each inverse just has +ℓ where E had −ℓ. Multiplied in this order, they don't interfere: every ℓ lands in its own position. No extra arithmetic. L is the receipt of every step you took.
Worked example
Why it matters
For n unknowns, factoring costs about n³/3 operations. Each solve after that costs about n². With n = 1000: around 330 million for the factoring, then about a million per new b.
So when the matrix stays the same and only the right side changes, as in a simulation stepping through time, you factor once and reuse it. The lab has lu(A) if you want to check your own matrices.