LU decomposition
LU decomposition writes a square matrix A as a product A = L*U, where L is lower-triangular (zeros above the diagonal) and U is upper-triangular (zeros below it). It is really just Gaussian elimination written down and remembered: U is the staircase form you reach by eliminating, and L records the multipliers you used to get there.
Why bother remembering it? Because solving A*x = b for one b is easy, but solving it for many different b is wasteful if you redo elimination each time. With L and U in hand, each new b is solved by two quick triangular sweeps (forward then backward substitution) instead of full elimination.
A small caveat for honesty: plain LU can stumble if a pivot lands on zero or something tiny, so practical software uses row swaps (this is called LU with partial pivoting, often written P*A = L*U). It is one of the workhorses behind everyday equation-solving.
L holds the elimination multiplier 1.5; U is the resulting upper-triangular staircase.
Once A = L*U is computed, each extra right-hand side b costs only two cheap triangular solves.