Is A-1 = adj(A) true if A is invertible and det(A) = 1?

[annoymage]

Homework Statement

Show that it is false

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

Homework Equations

N/A

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

A^-1 = adj(A)

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?

 

[Gregg]

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

 

[annoymage]

hmm, yea, that is a true. but i don't understand what you trying to say..

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

 

[Gregg]

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

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

 

[annoymage]

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

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?

 

[vela]

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.

 

[annoymage]

hmm, I am 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

 

[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?

 

[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?

 

[annoymage]

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

 

[vela]

which means, this statement

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

is always true, right?

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.

how can i prove this wrong?

Find an A that is invertible and whose inverse is its adjoint but for which det A is not equal to 1.

 

[annoymage]

i can't find any counter example. T_T

somebody help me

 

[vela]

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.