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Matrix Differentiated (Posted on 2019-05-24) Difficulty: 3 of 5
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).

No Solution Yet Submitted by Danish Ahmed Khan    
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clarifcation | Comment 1 of 3
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

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