pöördmatrix

A pöördmatrix, also known as an inverse matrix in English, is a fundamental concept in linear algebra. Given a square matrix *A* of size *n × n*, its inverse, denoted as *A⁻¹*, is another square matrix such that when it is multiplied by *A*, the result is the identity matrix *I* of the same size. Mathematically, this relationship is expressed as *A⁻¹A = AA⁻¹ = I*. Not all matrices possess an inverse; those that do are called invertible or non-singular matrices.

The existence of an inverse is determined by the matrix's determinant. A matrix has an inverse if and only if its determinant is non-zero. The inverse can be computed using various methods, including the adjugate method, Gaussian elimination, or computational algorithms optimized for efficiency. For a 2×2 matrix *A* with elements *a, b, c, d*, the inverse is given by the formula:

*A⁻¹ = (1/det(A)) * [d -b; -c a],*

where *det(A) = ad - bc*.

Inverses of matrices are widely used in solving systems of linear equations. If *Ax = b*, then multiplying both sides by *A⁻¹* yields *x = A⁻¹b*, providing a direct solution for *x*. This property is particularly useful in applications such as computer graphics, engineering, and data science, where transformations and computations rely on matrix operations.

The concept of matrix inversion extends to other mathematical structures, including generalized inverses for non-square or singular matrices, which have applications in least-squares solutions and statistical modeling. While inverses simplify certain calculations, their computation can be numerically unstable for large or ill-conditioned matrices, necessitating careful handling in practical implementations.