Question

Let A and B be n times n matrices. Show that if AB is non- singular, then A and B must be nonsingular.

Matrices
ANSWERED
asked 2020-11-22
Let A and B be \(n \times n\) matrices. Show that if AB is non- singular, then A and B must be nonsingular.

Answers (1)

2020-11-23
Step 1
We will first consider the given statement,
The objective is to show that if AB is non singular then A and B must be nonsingular.
Step 2
We will let \(v \in R^n\) and suppose that (AB)x=0
Here, we have to prove that x=0
Next , we will let that \(y=Bx \in R^n\)
Thus, we have,
Ay=A(Bx)
=(AB)x
=0
Step 3
As A is non-singular , this implies that y=0
Thus, we have , y=Bx=0
Also, B is non-singular , this implies that x=0(x)=0
This further implies that if
(AB)x=0(AB)x
then , we must have, x=0
=0
Thus, this means that AB is non-singular
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