Solution Found!
Let A be an m x n matrix with rank m. Prove that there exists an nxm matrix B such that
Chapter 3, Problem 21(choose chapter or problem)
Let A be an \(m \times n\) matrix with rank m. Prove that there exists an \(n \times m\) matrix B such that \(A B=I_{m}\).
Questions & Answers
QUESTION:
Let A be an \(m \times n\) matrix with rank m. Prove that there exists an \(n \times m\) matrix B such that \(A B=I_{m}\).
ANSWER:Step 1 of 3
We can see that the rank of the matrix \(A_{m \times n}\) is equal to the number of its rows. Therefore, we can conclude that \(n \geq m\) since the rank of a matrix can't exceed the number of its rows or columns. According to Theorem 3.6 , using a finite number of elementary row and column operations we can transform the matrix \(A_{m \times n}\) of rank m into the matrix
\(D_{m \times n}=\left(\begin{array}{cc}
I_{m} & O_{1} \\
O_{2} & O_{3}
\end{array}\right)\)