Let A and B be two n×n matrices with real entries.
Define the function f : R → R by
f(x) = det(A + Bx)
(i) Show that f(3)(x) = 3! det B.
(ii) Show that in general f(n)(x) = n! det B.
f(n)(x) is the nth derivative of f(x).
Danish Ahmed Kahn, I'm sorry for not noticing and bringing this up during review, but I am a bit confused about this problem.
If A,B are nxn matrices, then A+Bx is also an nxn matrix where each element is linear in terms of x. Thus det(A+Bx) is a nth degree polynomial in x and thus the kth derivative would be a (n-k)th polynomial in x. Perhaps I am missing something but the value you show for the derivatives has no x in it.
Edited on May 24, 2019, 2:09 pm
Edited on May 24, 2019, 2:10 pm
Posted by Daniel
on 2019-05-24 14:09:25