Question

Let A be an \(m \times n\) matrix of rank \(r\). Prove that there is a nonsingular \(m \times m\) matrix \(P\) and a nonsingular \(n \times n\) matrix \(Q\) such that the matrix \(B=P A Q=\left(b_{i j}\right)\) has entries \(b_{i i}=1\) for \(1 \leq i \leq r\) and all other entries \(b_{i j}=0\)

Answer 1

Let A be m times n matrix of rank r To prove: There is a non singular matrix (m times) P and a non singular matrix Q such that the matrix B = PAQ = (b_ij) has entries b_ij = 1 for 1 lessthanorequalto i lessthanorequalto r and all other entries b_ij = 0 We need to prove that if A can be reduced to two matrices P and Q bot n of the above form where the numbers of non - zero elements are r and r' respectively, by different sequence of elementary Operations, then r = r' and so P = Q rightarrow The above matrix B = PAQ = [I_r 0 0 I_r] in the canonical form for equivalence which is same as we want to show rightarrow We outline the proof that reduction is possible to prove that we always get the same values of r we need a different argument The proof is by