Segédmátrix

Segédmátrix, often translated as auxiliary matrix or cofactor matrix, is a concept in linear algebra related to square matrices. For a given square matrix A, its segédmátrix is a matrix of the same dimensions where each element is the determinant of a submatrix formed by deleting the corresponding row and column from A. This submatrix determinant is then multiplied by (-1)^(i+j), where i is the row index and j is the column index of the element in the original matrix.

The segédmátrix plays a crucial role in the calculation of the adjugate matrix (also known as the adjoint matrix) and the inverse of a matrix. The adjugate matrix is the transpose of the segédmátrix. If a matrix A is invertible, its inverse A⁻¹ can be found by dividing the adjugate matrix of A by the determinant of A. Therefore, understanding and calculating the segédmátrix is a fundamental step in these matrix operations. The notation for the element in the i-th row and j-th column of the segédmátrix is often denoted as C_ij. The calculation involves minors, which are the determinants of these submatrices before the alternating sign is applied.