This post is dedicated to some important properties regarding adjoint of matrix. If, you want to go through their proves then click particular property. Note that these properties are only valid for square matrices as adjoint is only valid for square matrices.
where, A is a square matrix, I is an identity matrix of same order as of A and
represents determinant of matrix A.
determinant of adjoint A is equal to determinant of A power n-1 where A is invertible n x n square matrix.
{A is n x n invertible square matrix}
You can also take examples to verify these properties.