# Inverse of a Matrix

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Inverse of a Matrix

For a given square matrix *A* = ǀǀ*a _{ij}*ǀǀ

^{n}

_{1}of order

*n*there exists a matrix

*B*= ǀǀ

*b*ǀǀ

_{ij}^{n}

_{1}of the same order (called inverse matrix) such that

*AB*=

*E*, where

*E*is the unit matrix; then the equation

*BA*=

*E*also holds. The inverse of a matrix

*A*is designated as

*A*

^{–1}. For the existence of the inverse of a matrix

*A*

^{–1}, it is necessary and sufficient that the determinant of the given matrix

*A*be nonzero; that is, the matrix

*A*must be nonsingular. The elements

*b*of the inverse of a matrix are found by the formula

_{ij}*b*=

_{ij}*A*/

_{ji}*D*, where

*A*is the cofactor of the element

_{ji}*a*of matrix

_{ij}*A*and

*D*is the determinant of matrix

*A*.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.