A useful identitiy in circuit theory (Q1338647)

A useful identity for circuit theory is obtained. The authors considered the linear system \(AX= B\), where \(A\) is a square matrix and \(X\) and \(B\) are column matrices. If \(B\) remains the same but an element of \(A\), say \(a_{ij}\), is modified, then it is asked to find the derivative of the perturbed solution \(X'\) with respect to the modified element. The result can be proved to be of much interest in sensitivity theory, Kron's method of tearing and \textit{F. H. Branin's} formulae [The relation between Kron's method and the classical methods of network analysis. Matrix Tensor Q. 12, 69-105 (1962)].