Let A and B be n x n matrices,
assume AB is invertible and show that both A and B are invertible.
what?
AB is invertible ----->
There exists a matrix X such that:
(AB)X = 1
But (AB)X = A(BX). So,
AY = 1
for Y = BX
Also:
X(AB) = 1
And thus
ZB = 1
for Z = XA
To show that both A and B are invertible given that AB is invertible, you need to show that there exist inverse matrices for both A and B.
First, you know that AB is invertible, meaning there exists a matrix X such that (AB)X = I, where I is the identity matrix.
But (AB)X = A(BX) since matrix multiplication is associative. Therefore, A(BX) = I.
This means that there exists a matrix Y such that AY = I, where Y = BX.
Similarly, since (AB)X = I, you can also write X(AB) = I. Rearranging this equation gives ZB = I, where Z = XA.
Therefore, you have shown that both A and B have inverse matrices. A's inverse is Y, and B's inverse is Z.
So, to summarize, if AB is invertible, you can find inverse matrices Y and Z such that AY = I and ZB = I, respectively. Hence, both A and B are invertible.