Matrix inverse

The inverse of a square matrix is a matrix that multiplies with it, on either side, to produce the identity matrix. It exists exactly when the original matrix is nonsingular.

Identity matrix: A square matrix with ones on its main diagonal and zeros elsewhere; it leaves matrices unchanged under multiplication. It is the neutral element that the inverse must produce.
Matrix multiplication: An operation that combines matrices to represent composition of linear transformations or multiplication of linear maps. The inverse is defined through multiplication with the original matrix.
Change of basis: A conversion between coordinate descriptions of the same vector using different bases. Inverse change-of-basis matrices convert coordinates back to the original basis.
Moore–Penrose pseudoinverse: A uniquely defined generalized inverse that applies to rectangular and singular matrices and supports least-squares solutions. It extends inversion to matrices for which an ordinary inverse does not exist.
Determinant: A scalar associated with a square matrix that encodes properties including invertibility and signed volume scaling. A matrix has an inverse exactly when its determinant is nonzero.
Square matrix: A matrix with the same number of rows and columns. Only square matrices have ordinary two-sided inverses.
Matrix decomposition: A factorization of a matrix into simpler matrices with useful computational or structural properties. Factoring a matrix is usually more stable than explicitly forming its inverse.
Matrix condition number: A measure of how strongly a matrix can amplify relative errors in inputs or computations. An inverse may exist but be numerically unreliable when the condition number is large.
Gaussian elimination: A method for solving linear systems by applying row operations to simplify their coefficient matrix. Row reduction can compute an inverse by transforming the matrix into the identity.
Linear transformation: A function between vector spaces that preserves vector addition and scalar multiplication. A matrix inverse represents reversing the transformation encoded by a matrix.

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