Linear algebra: matrices and vector spaces

Matrices organise systems of equations, and economics is full of systems: input-output tables, simultaneous markets, portfolio weights. A matrix is a rectangular array; multiplying two matrices combines their linear maps, and the inverse of a square matrix (when it exists) solves the system in one step. Worked example: the system x + 2y = 8 and 3x + 4y = 18 has coefficient matrix with determinant 1 x 4 - 2 x 3 = -2, so it has the unique solution x = 2, y = 3 (check: 2 + 6 = 8 and 6 + 12 = 18).