Let’s look at some identities of determinant

Determinant of Identity matrix = 1
det (I) = 1
Where I is identity matrix of any order (1 × 1, 2 × 2, 3 × 3, … n × n)

det (A
^{
T
}
) = det A
A ^{ T }  = A

AB = A B

A
^{
−1
}
 = 1/(A)
Proof :
AA ^{ −1 } = I
AA ^{ −1 }  = I
A A ^{ −1 }  = −1
A ^{ −1 } = 1/ A

kA = k ^{ n } A
where n is order of matrix

Similarly,
−A = −1 × A
= (−1) ^{ n } × A 
(adj A) A = A (adj) = A I

Deteminant of adj A
We know that
A (adj A) = AI
Taking determinant both sides
A (adj A)  = AI 
We know that
kA = k ^{ n } A
 A  adj A  =  A  ^{ 𝑛 } I
 A  adj A  =  A  𝑛 × 1
 adj A  = 𝐴 ^{ 𝑛 } / 𝐴
adj A  =  A  ^{ 𝑛−1 }
 adj A  =  A  ^{ n1 }
where n is the order of determinant
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