Why is the value of the determinant of adjoint of a matrix not equal to 1 ?

Question

What is the value of determinant of adjoint( A). What I am not getting in this is that since $AA\ ^{-1}=\mathbb{I}_{n}$ .Now if I take determinant on both sides I will get $\mid A\mid \mid A^{-1}\mid=\mathbb{I}_{n}=1$  ,now $\mid A^{-1}\mid=1/\mid A\mid$ ,then $\mid A^{-1}\mid=\mid Adjoint A\mid/ \mid A\mid$. By substituting the above value I get $\mid Adj A\mid =1$. What is wrong in this statement ? since value of $\text{ det(adj(A))}=det(A)^{n-1}$

Comments

in your approach also I got the same answer

you did wrong here

∣A⁻¹∣=∣AdjointA∣/∣A∣

it should be

∣A⁻¹∣=∣ AdjointA / ∣A∣ |

Answer 1

A A^-1  = I_n

A (adj A) = |A| I_n

take determinants Both side

| A (adj A) | = |  |A| I_n |

==>We know that if A be an n-rowed square matrix  and ' k'  be any scalar then

| K A | = Kⁿ |A|

Now

| A | | adj A| = | A |ⁿ | I_n |

| A | | adj A| = | A |ⁿ

| adj A| = | A |^n-1

Comments

I have gone through this proof ,I just wanted to ask what is wrong in my approach ?

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∣A^-1∣=|adj(A) / ∣A∣|

        =|adj(A)| / |A|ⁿ

and not |adj(A)| / ∣A|