Question

Can anyone help me with this proof? Am I on the right track?
Let A and B be equivalent square matrices. Prove that A is nonsingular if and only if B is nonsingular

Answer 1

Given: Matrix A and Matrix B are equivalent square matrices.
Since we are given a biconditional statement, we need to prove two situations:
If Matrix A is nonsingular then Matrix B is nonsingular
If Matrix B is nonsingular then Matrix A is nonsingular

Answer 2

Let's see....
Theorem 2.13 states that two m x n matrices A and B are equivalent of and only if B = PAQ for some nonsingular matrices P and Q.

Answer 3

We have a product of nonsingular matrices P and Q. If B is nonsingular, B=PAQ, then A is nonsingular A=P^-1BQ^-1

Answer 4

Conversely, if A is nonsingular, A = PBQ, then B is nonsingular, B=P^-1AQ^-1 because P and Q are products of nonsingular matrices.