Questions about matrix

(1)Why we define adj A where A is a matrix?

Since the inverse A=adj A/det A, we can simplify these equation by making a new definition of adj A.

(2) Prove that Inverse of A=adj A/det A

We need to show A * (adj A/det A) = (adj A/det A) * A = Identity Matrix.

Let A be a n x n matrix and let cij be the cofactors of aij , the adjoint A,
denoted as Adj A, is defined as :

Adj A = [cij]

The reason Adj A is so important because

(1) A . Adj A = Adj A * A = det(A) . I (by doing multiplication from left and from right)
In particular, if A is nonsingular, then inverse of A = Adj A/ det(A)