For finding the inverse of a matrix A, we convert the expression A = I A (where I is identity matrix), such that we get I = B A ( here B is inverse of matrix A) by employing elementary row or column operations. But why do these operations work? Why does changing elements of a complete row by another corresponding row work but e.g. we can't add/subtract the same number in the equality? Is there some proof of these operations? What is the principle/intuition behind these operations? Can they be explained by any first principles?