Matrix Algebra Summary

Linear Algebra (Matrices) · Axiom Academy

Unit 3 wrap-up: the matrix inverse, determinants, the test for invertibility, and Cramer's rule — all in one place. The inverse A^ -1 is the matrix that undoes A : AA^ -1 =A^ -1 A=I . A square matrix is invertible if and only if — one number decides everything. The determinant is a scaling factor : is how much A stretches area (2D) or volume (3D); a sign flip means orientation reverses. Pick the method to fit the matrix: the formula for small ones, the augmented row-reduction for any size, cofactor expansion for determinants, Cramer's rule for one variable. Determinants are multiplicative : , which makes many results fall out for free. Core Concept The Matrix Inverse For a square matrix A , its inverse A^ -1 is the unique matrix that returns the identity when multiplied on either side. Geometrically, A^ -1 reverses the transformation A performs. Exists when: (square matrices only). Watch out for: if an inverse exists it is unique — there is never more than one. Core Concept The Inverse Formula Swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant. A fast mental computation whenever . When to use: matrices — quick by hand. Watch out for: if ad-bc=0 the matrix is singular; there is no inverse. Core Concept The Augmented-Matrix Method Augment A with the identity, then row-reduce until the left block becomes I — whatever lands on the right is A^ -1 . The general-purpose method that works at any size.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.