# Linear Algebra Proof using Inverses

1. Feb 18, 2015

### B18

1. The problem statement, all variables and given/known data
Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA.
2. Relevant equations

3. 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?

2. Feb 18, 2015

### LCKurtz

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

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

3. Feb 18, 2015

### 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.

4. Feb 18, 2015

### 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

5. Feb 18, 2015

### 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.

6. Feb 18, 2015

### 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

7. Feb 18, 2015

### RUber

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

8. Feb 18, 2015

### 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.

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

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

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.

Last edited: Feb 18, 2015