# Proving a Linear System

1. Apr 12, 2010

### annoymage

1. The problem statement, all variables and given/known data

Show that it is false

If A is invertible and A-1 = adj A, then det A= 1

2. Relevant equations

N/A

3. The attempt at a solution

------------------------------------------------------
A is invertible iff A-1A = I

implies det(A-1A) = det(I)
implies det(A-1)det(A) = 1
implies det(A) = 1 or -1--------------------------------this is wrong, just ignore this part (edited)
---------------------------------------------------------
AND

which is false because

A-1 = $$\frac{1}{det(A)}$$ adj(A)

so, which means the premise is already false

false implies (true or false)

is a true statement

so, is it the question wrong, or i did any mistake?

Last edited: Apr 12, 2010
2. Apr 12, 2010

### Gregg

$$\Delta = |A| \Rightarrow |A^{-1}|=\frac{1}{\Delta}$$

3. Apr 12, 2010

### annoymage

hmm, yea, that is a true. but i dont understand what you trying to say..

sorry if i have a bad english or grammar.. hoho

4. Apr 12, 2010

### Gregg

$$|A||A^{-1}|=I\Rightarrow |A^{-1}|=\frac{1}{|A|}$$

Not true:
$$|A||A^{-1}|=I\Rightarrow |A|=\pm 1$$

5. Apr 12, 2010

### annoymage

yeaaaaaaaaa, this is wrong, owho sorry,

but then,

still the premises are false...

True and False (implies) True or False

is a true statement..

False and False (implies) True or False

is also true statement..

So, which means, the question about, "prove that this is false" is incorrect?

6. Apr 12, 2010

### vela

Staff Emeritus
When you're proving a statement, you assume the premises are true and show the conclusion then follows.

What you're thinking of regarding the implication is that if you have some matrix A for which the premises don't hold, then the implication is true because F->T is true. However, if you have a matrix A for which the premises hold (for example, I=A=A-1=adj(A)), then the conclusion must also be true if the implication is to be valid.

If you're trying to show the implication is invalid, you need to show that the conclusion doesn't necessarily follow even if the premise is true, i.e. show T->F. Typically, you do this by finding a counterexample.

7. Apr 12, 2010

### annoymage

hmm, im sorry if i get the wrong meaning of what you are saying

but the question ask to prove that
"If A is invertible and A-1 = adj A, then det A= 1"
is wrong

If i assume that "If A is invertible and A-1 = adj A" is true, then i can proof this is false by contradiction..

which also means it is a false
and whatever conclusion you get, you still have a true statement (F->T or F)

sorry for my bad english.. :P

8. Apr 12, 2010

### annoymage

which means, this statement

"A is invertible and A-1 = adj A, then det A= 1"

is always true, right?

how can i prove this wrong?

9. Apr 12, 2010

### xaos

"A is invertible iff A-1A = I

implies det(A-1A) = det(I)
implies det(A-1)det(A) = 1
implies det(A) = 1 or -1"

under what conditions would it be true (edit:possible) that detA=-1?

10. Apr 12, 2010

### annoymage

no no, that's wrong, i made mistake with that..

11. Apr 12, 2010

### vela

Staff Emeritus
No, if it is always true, you can't prove it wrong. If the statement is wrong, there must be a case where the premise is true but the conclusion is false.
Find an A that is invertible and whose inverse is its adjoint but for which det A is not equal to 1.

12. Apr 12, 2010

### annoymage

i can't find any counter example. T_T

somebody help me

13. Apr 12, 2010

### vela

Staff Emeritus
I'm afraid you're just going to have to think about it. Stick with simple matrices (so it's obvious what the adjoint, inverse, and determinant are). It's pretty easy to come up with a counterexample.

If you're still stuck, one tactic you can take is to try prove the incorrect statement. Hopefully, you'll run into a roadblock which will give you a hint as to what a counterexample is.