I would like to know if there exist any identity regarding to the inverse of a matrix in which a row or (or a column) has been multiplied by a constant.
I know that if whole matrix is multiplied, inverse matrix is inverse of the original multiplied by the inverse of the constant.
Multiplication of a column of matrix by a factor is the multiplication by a diagonal matrix. For example if you want to multiply the first column of a matrix by $c\in \mathbb R$ you have to multiply from the right by the matrix $$ C:=\left( \begin{array} &c&0&\ldots &0\\ 0&1&\ldots&0\\ 0&0&\ddots&0\\ 0&0&\ldots&1 \end{array}\right) $$ now for any matrix $A$ we have $$(AC)^{-1}=C^{-1}A^{-1}.$$ Calculate $C^{-1}$ and you have a formula for the inverse of the product.