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

floymdiT
2020-11-22
Answered

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

You can still ask an expert for help

faldduE

Answered 2020-11-23
Author has **109** answers

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

Here, we have to prove that x=0

Next , we will let that

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

Jeffrey Jordon

Answered 2022-01-27
Author has **2262** answers

Answer is given below (on video)

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