finding suitable column matrix

Question

I have a row matrix A:1*n and need to find suitable column matrix B:n*1 to multiplying existing row matrix BA=C to reach inversible square matrix C:nn ?

Answer 1

Hint:

For $$ A=[a_1,a_2, \cdots ,a_n] \quad B=[b_1,b_2, \cdots ,b_n]^T $$ we have: $$ BA= \begin{bmatrix} b_1\\ b_2\\ .\\ b_n \end{bmatrix} \begin{bmatrix} a_1,a_2,\cdots,a_n \end{bmatrix}= \begin{bmatrix} b_1a_1&b_1a_2\cdots b_1a_n\\ b_2a_1&b_2a_2\cdots b_2a_n\\ .\\ b_na_1&b_na_2\cdots b_na_n\\ \end{bmatrix} $$

so the columns of $BA$ are clearly not lineraly independent.