Prove there exists a matrix that satisfies the equation

Question

Let A be an mxn matrix with rank m. Prove that there exists an nxm matrix B such that AB = $I_{m}$.

-Since the rank of A is m, and A is an mxn matrix, the matrix must have full row rank, so there must also exist some $b_{j}$ that satisfies the equation A$b_{j}$= $I_{j}$, where j stands for the jth column of a matrix.

Answer 1

"-Since the rank of A is m, and A is an mxn matrix, the matrix must have full row rank, so there must also exist some $b_{j}$ that satisfies the equation A$b_{j}$= $I_{j}$, where j stands for the jth column of a matrix."

You're almost/pretty-much done. You could use the following fact to finish your proof. Suppose that we have $AB = C$ and $b_j,c_j$ are the $j$-th columns of $B$ and $C$, respectively. Then in general, we have that $c_j = Ab_j$.