Proving the complexity of recursive determinant

Question

I want to prove why Laplace determinant or recursive algorithm complexity is n!. Can anyone prove it for me? I don't know how could it then be n!, given that the equation T(n)=nT(n-1)+3n-1 only involves multiplication and addition.

Answer 1

Try to expand:

T(n) = n T(n-1) + 3n-1 = 
       n ((n-1)T(n-2) + 3(n-1)-1) + 3n-1 =
       n (n-1) T(n-2) + 3 n (n-1)  - n + (3n - 1)

Now by induction you can show that (if T(1) = 1):

T(n) = n (n-1) (n-2) ... 1 + 3(n + n (n-1) + ... + n!) - 
      (1 + n + n (n-1) + ... + n (n-1) ... n * (n-1) * ... * (n-2)) 
     = Theta(n!)