Linear Algebra Proof using Inverses

[B18]

Homework Statement

Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B⁻¹ = CA.

Homework Equations

The Attempt at a Solution

I think I have this figured out, just checking it. Heres what I got:
ABC=I
(ABC)B^-1=IB^-1
(B*B^-1)AC=IB^-1
I*AC=IB^-1 Cancel I using left hand cancellation property
AC=B^-1
Thus B^-1=CA

Is every thing I've done here mathematically correct?

 

[LCKurtz]

Homework Statement

Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B⁻¹ = CA.

Homework Equations

The Attempt at a Solution

I think I have this figured out, just checking it. Heres what I got:
ABC=I
(ABC)B^-1=IB^-1

How do you know $B$ has an inverse to use? You are trying to prove that.

(B*B^-1)AC=IB^-1

And, even if you did, how did you get that step? Matrix multiplication isn't commutative.

 

[B18]

Ok, yes I see what you're saying. I can't do the steps I did there. I know that B has to have an inverse because A,B, and Care square matrices and their product is the identity matrix.

 

[B18]

Is this a correct path to go down on this proof?
We have ABC=I
(AB)C=I. Since (AB)C=I we know that (AB) and C are both invertible. Also this tells us that C=(AB)^-1, and (AB)=C^-1

 

[RUber]

You can also reorder the multiplication using
CABC = CI = C =IC
Implies CAB = I.
Same logic as in your last post should bring you to the solution you are looking for.

 

[B18]

How does this look:
We have ABC=I
C(ABC)=CI
CABC=C
(CABC)A=CA
(CAB)CA=CA This implies that CAB=I
CA(BCA)=CA This implies that BCA=I
CAB=BCA
(CA)B=B(CA) Then B must be invertible
Therefore BCA=I
CA=B^-1

 

[RUber]

Looks good to me. You hit all the important points.

 

[LCKurtz]

I think you need to flesh out your argument with a few more details. Your steps may be correct, but if this is a homework problem you need to fill in some reasons.

How does this look:
We have ABC=I
C(ABC)=CI
CABC=C
(CABC)A=CA
(CAB)CA=CA This implies that CAB=I
CA(BCA)=CA This implies that BCA=I

Why do those imply those?

CAB=BCA
(CA)B=B(CA) Then B must be invertible

Why must B be invertible? That statement by itself doesn't imply it.

Therefore BCA=I

Why the "therefore" now? Didn't you already have BCA=I above?

CA=B^-1

Like I said above, your statements may be true, but your teacher is going to want to know if you know why they are true.