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MATH 217 – LINEAR ALGEBRA
HOMEWORK 10, DUE Wednesday, December 1, 2021 at 11:59 pm
Submit Part A and Part B as two separate assignments. Include the following information in the top
left corner of every assignment:
your full name,
instructor’s last name and section number,
homework number,
whether they are Part A problems or Part B problems.
A few words about solution writing:
? Unless you are explicitly told otherwise for a particular problem, you are always
expected to show your work and to give justification for your answers. In particular,
when asked if a statement is true or false, you will need to explain why it is true or false to
receive full credit.
Write down your solutions in full, as if you were writing them for another student in the class
to read and understand.
Don’t be sloppy, since your solutions will be judged on precision and completeness and not merely
on basically getting it right.
Cite every theorem or fact from the book that you are using (e.g. By Theorem 1.10 . . .).
If you compute something by observation, say so and make sure that your imaginary fellow
student who is reading your proof can also clearly see what you are claiming.
Justify each step in writing and leave nothing to the imagination.
Part A (10 points)
Solve the following problems from the book:
Section 6.3: 10, 18
Section 7.1: 7, 39, 47
Section 7.2: 15, 33
Part B (25 points)
Note: throughout this homework set, eigenvalue means real (as opposed to complex) eigenvalue and
diagonalizable means diagonalizable over R (as opposed to over C). (If you do not yet understand
what this means, that’s ok, and you can ignore this note.)
Problem 1. In each part below, determine whether or not the given statement is true for all n × n
matrices A and B, and justify your answer with a proof or counterexample.
(a) If A is diagonalizable then A> is diagonalizable.
(b) If A is diagonalizable and invertible, then A?1 is diagonalizable.
(c) If A and B are diagonalizable, then A+B is diagonalizable.
(d) If A and B are diagonalizable, then AB is diagonalizable.
(e) If A and B are similar matrices and A is diagonalizable, then B is diagonalizable.
(f) If A2 is diagonalizable, then A is diagonalizable.
(g) If A is diagonalizable, then A2 is diagonalizable.
(h) If A is diagonalizable, then A2 + A is diagonalizable.
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MATH 217 – LINEAR ALGEBRA HOMEWORK 10, DUE Wednesday, December 1, 2021 at 11:59 pm 2
Problem 2. A polygon P ? R2 is called a lattice polygon if all of its vertices have integer coordinates.
A half-integer is a number of the form
1
2
n where n ∈ Z (so all integers are also half-integers, but so are
numbers like.
(a) Prove that the area of a lattice triangle is always a half-integer. [Hint: use determinants.]
For part (b), you may freely use the following lemma from geometry (see Figure 1 below): any n-sided
polygon in R2 can be broken up into n 2 non-overlapping triangles by drawing in n 3 chords.
(b) Prove that the area of any lattice polygon is always a half-integer.
Figure 1. Triangulating a polygon.
Problem 3. We say that two n× n matrices A and B commute if AB = BA. Recall also that a scalar
matrix is a matrix of the form A = λIn for some λ ∈ R and n ∈ N.
(a) Prove that if A is a scalar matrix, then A commutes with every n× n matrix B.
(b) Assuming D is a diagonal matrix with distinct eigenvalues, find all matrices that commute with D.
(c) Prove the converse of (a): if A is an n× n matrix that commutes with every other n× n matrix B,
then A is a scalar matrix.
(d) Prove that if the n × n matrices A and B commute and ~v is an eigenvector of A, then B~v is also
an eigenvector of A.
(e) Suppose thatM and N are diagonalizable matrices, each having distinct eigenvalues, that commute
with each other. Prove that M and N are simultaneously diagonalizable, meaning that there exists
a single matrix S such that SMS?1 and SNS?1 are both diagonal.
[Hint: use your results from parts (a) (d).]
Problem 4. Recall that every orthogonal transformation of R2 is either rotation by some angle about
the origin or else reflection across a line through the origin. We will now characterize all orthogonal
transformations of R3. Throughout this problem, let A be a 3× 3 orthogonal matrix.
(a) Prove that | det(A)| = 1.
(b) Prove that A has an eigenvector ~v with corresponding eigenvalue 1 or 1. [Hint: you may use the
fact from calculus that every polynomial of odd degree has a real root.]
(c) Prove that if ~v is an eigenvector of A with eigenvalue 1, then A is either rotation by some angle
around the line ` = span(~v) or else reflection through a plane containing ~v. [Hint: Look at what A
does on the subspace ⊥. In particular, note that A~x ∈ ⊥ for every ~x ∈ `⊥.]
(d) Prove that if 1 is not an eigenvalue of A, then A is the transformation obtained by first rotating
around some line through the origin, and then reflecting across the plane ⊥.
(e) Combine the previous parts to give a complete description of all orthogonal transformations of R3.