Question

For any matrix A prove that A( adj A) = |A| × I_n

Answer 1

I assume that A is a square matrix, then we know
The inverse of A = adj(A) / det(A) where det is the determinant
multiply both sides of the = by A and we get
A*inverse of A = (A*adj(A)) / det(A) and A*inverse of A = (adj(A)*A) / det(A)
note that * means multiply
the above implies that
I = (A*adj(A)) / det(A) and I = (adj(A)*A) / det(A)
From above, we can say that det(A)I = A*adj(A) and det(A)I = adj(A)*A, then
A*adj(A) = adj(A)*A= det(A)*I