Problem Set 6 (due Friday October 15)
MATH 110: Linear Algebra
Each problem is worth 5 points.
PART 1
1. Curtis p. 150 2.
2. Curtis p. 150 6.
3. Curtis p. 182 1.
4. Curtis p. 182 2.
Remember that the starred problem is non collaborative.
Problem 1(10)
Two matrices Aand Bare said to be similar if there exists a matrix C
such that B=C−1AC. Show that two matrices that are similar have the
same determinant.
Problem 2(10)
For each of the following statements about square matrices, give a proof
or find a counter example:
a) det(A+B) = det(A) + det(B).
b) det((A+B)2) = (det(A+B))2.
c) det((A+B)2) = det(A2+ 2AB +B2).
d) det((A+B)2) = det(A2+B2).
Problem 3 (15)
Given n2functions fij on an interval (a, b) define F(x) = det[fij(x)] for
each xin (a, b). Prove that the derivative F0(x) is a sum of ndeterminants
F0(x) =
n
X
i=1
detAi(x).
where Ai(x) is the matrix obtained by differentiating the functions in the ith
row of [fij(x)].
Problem 4 (10)
Prove Demoivre’s theorem:
(cos θ+isin θ)n= cos nθ +isin nθ.
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