Page 185 - 35Linear Algebra

8.3 Review Problems 185

express your results in terms of tr(A)? What about the first order term
in det(A + tI n ) for any arbitrary n × n matrix A in terms of tr(A)?
Note that the result of det(A + tI 2 ) is a polynomial in the variable t
known as the characteristic polynomial.

13. (Directional) Derivative of the determinant:
n
n
Notice that det: M → R (where M is the vector space of all n × n
n n
2
matrices) det is a function of n variables so we can take directional
derivatives of det.
Let A be an arbitrary n × n matrix, and for all i and j compute the
following:

(a)
i
det(I 2 + te ) − det(I 2 )
j
lim
t→0 t
(b)
i
det(I 3 + te ) − det(I 3 )
j
lim
t→0 t
(c)
i
det(I n + te ) − det(I n )
j
lim
t→0 t
(d)
det(I n + At) − det(I n )
lim
t→0 t
i
Note, these are the directional derivative in the e and A directions.
j
14. How many functions are in the set

{f : {1, . . . , n} → {1, . . . , n}|f −1 exists} ?

What about the set
{1, . . . , n} {1,...,n} ?

Which of these two sets correspond to the set of all permutations of n
objects?

185