On the other hand, assuming that our original matrix A is invertible, this matrix X 1 {\displaystyle X_{1}} has columns A − 1 b , A − 1 v 2 , … , A − 1 v n {\displaystyle A^{-1}b,A^{-1}v_{2},\ldots ,A^{-1}v_{n}} , where v n {\displaystyle v_{n}} is the n-th column of the matrix A. Recall that the matrix A 1 {\displaystyle A_{1}} has columns b , v 2 , … , v n {\displaystyle b,v_{2},\ldots ,v_{n}} . Hence we have