How to get the linear combination of the rows that result in Identity matrix?

Question

I have a matrix $B$ whose rows are $n$ and whose columns are $k$. The rank of the matrix is $k$. In convex optimization book (by Stephen Boyd) it is written that in this case we can find a non-singular matrix $A$ such that $$AB= \left[ {\begin{array}{cc} A_1\\ A_2\\ \end{array} } \right]B=\left[ {\begin{array}{cc} I\\ 0\\ \end{array} } \right] \ $$ Now I know that rows of $A_2$ are the basis of the left null space of $B$. But I do not know how to find the rows of $A_1$ and how they relate to the row space of $B$ (if they do). Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

One possible way to find such a matrix $A$ is to perform elementary row operations to reduce $B$ to its RREF which is $\begin{bmatrix} I \\ 0 \end{bmatrix}$

Hence we can find elementary matrices such that

$$E_l \ldots E_1 B = \begin{bmatrix} I \\ 0 \end{bmatrix}$$

Hence we let $A= E_l \ldots E_1$ and $A_1$ is the first $k$ rows of $A$.