Matrix Inverse & Properties

Linear Algebra for Machine Learning · Axiom Academy

Definition, Properties & When It Exists The inverse of a matrix A, written A⁻¹, is a matrix that "undoes" the multiplication by A. A matrix A⁻¹ is the inverse of A if: Not all matrices have inverses—only square, non-singular matrices do. For a 2×2 matrix, there's a simple formula: A matrix A is invertible if and only if: A is square (same number of rows and columns) det(A) ≠ 0 (A is non-singular) A has rank n (full rank, where n is the size) Rows/columns are linearly independent The inverse of the inverse is the original matrix. The inverse of a product reverses the order! The inverse of the transpose equals the transpose of the inverse. Property 5: det(A⁻¹) = 1/det(A) The determinant of the inverse is the reciprocal. Matrix form of a system: Ax = b To solve, multiply both sides by A⁻¹: Important: This only works if A is square and invertible. For non-square systems, use least squares or other methods. Singular Matrix: A square matrix with det(A) = 0. It does NOT have an inverse. Its rows/columns are linearly dependent. Non-Singular Matrix: A square matrix with det(A) ≠ 0. It HAS an inverse. Its rows/columns are linearly independent. 3×3+: Gaussian elimination, adjugate method, or numerical algorithms Orthogonal: A⁻¹ = Aᵀ (instant!) Diagonal: Just invert diagonal entries

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.