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

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