Linear Algebra — Matrices
Matrices as machines, the determinant as a volume factor, inverses, and Cramer’s rule solving real systems.
A matrix is a machine for many numbers at once — it rotates, stretches, and shears whole spaces in one multiplication. Engineering runs on them: circuit analysis, structures, graphics, ML. The engine works with real matrices below.
- 01 Matrices & the determinant Enter a matrix with rows separated by ;. Its determinant is the single most informative number about it: the factor by which the machine scales area.…
- 02 The inverse — undoing the machine M⁻¹ undoes M, exactly like 1/x undoes multiplication. It exists precisely when det ≠ 0. The check verifies the defining property on an entry of M·M⁻¹ = I:
- 03 Solving systems — Cramer’s rule The Algebra 1 system 2x + y = 8, x − y = 1 in matrix form is A·[x, y] = [8, 1]. Cramer's rule solves it with determinants alone: replace a column with the…
- 04 Transpose & symmetry Where next: Differential Equations — where matrices and calculus join forces to describe anything that evolves.
next course: Differential Equations →