Pleeeeaaseeee proofs in linear algebra

[cleopatra]

Homework Statement

1) (AB)^-1=A^-1B^-1 A and B are nonsingular nxn
2) If A is nonsingular then A^-1 is nonsingular also and then (A^-1)^-1=A

The Attempt at a Solution

1) I do know that I have to multiply but I don´t know why. Can you tell me why??
This is how I do it:
((AB)^-1)*(A^-1B^-1 ))A(BB^-1)A^-1=A*In*A^-1=A*A^-1=In

2) ((A^-1)^-1)*A = ?

 

[tiny-tim]

Hi cleopatra!

(try using the X² tag just above the Reply box )

1) (AB)^-1=A^-1B^-1 A and B are nonsingular nxn

Your question is wrong … it should be prove (AB)^-1 = B^-1A^-1

 

[cleopatra]

ohh I see that now. Thanks :)

 

[cleopatra]

can you maby help me?

 

[tiny-tim]

well, to prove (AB)^-1 = B^-1A^-1 …

what is the definition of (AB)^-1 ?

 

[cleopatra]

the definition of (AB)-1 is
that AB is the inverce of AB

?

 

[tiny-tim]

the definition of (AB)-1 is
that AB is the inverce of AB

?

Yes , but what does that mean?

 

[cleopatra]

it means that when I multiply it with AB I get In
Right?

 

[tiny-tim]

it means that when I multiply it with AB I get In
Right?

Right!

And that's the way you answer any question …

ask yourself what the definition is, and then prove that the definition fits …

in this case, prove that when you multiply the answer by AB, you get I_n