Math 317: Linear Algebra
Practice Final Exam
Fall 2015
Name:
This practice final emphasizes the latest material covered in class. However, recall that
the final exam is comprehensive. To best prepare for the final, in addition to looking at
this practice final, look over all practice exams and real exams given in class, along with
the final exam topics sheet.
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1. Let A =
. Calculate Ak for all k ≥ 1.
1 −2
2. Let A and B be similar matrices.
(a) Prove that A and B have the same characteristic polynomial.
(b) Prove that A and B have the same eigenvalues.
3. (a) Let A be a n × n matrix with n distinct eigenvalues. Prove that det A is the
product of its eigenvalues.
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2
(b) Suppose that A is a symmetric matrix with real eigenvalues and A
=
.
1
2
Find A if det A = 6.
4. Prove that if λ1 , . . . , λk are distinct eigenvalues for A corresponding eigenvectors
v1 , . . . , vk , then the set {v1 , . . . , vk } is linearly independent.
5. Let V = {x ∈ R3 : −x1 + x2 + x3 = 0} and let T : R3 → R3 be the linear transformation that projects vectors onto V .
(a) Find an orthonormal basis B = {q1 , q2 , q3 } for R3 so that q1 , q2 span V and
q3 is orthogonal to V .
(b) Let C be the matrix of T with respect to B. Find C.
(c) Let A be the standard matrix A of T . Find A.
(d) Find the eigenvalues of C. Find the corresponding eigenvectors of C.
(e) Find the eigenvalues of A.
6. Suppose that A is an n × n matrix.
(a) Show by counterexample that Ax = 0, x 6= 0 does not necessarily imply that
A = 0.
(b) Prove that if Ax = 0 for all x 6= 0, then A = 0.
(c) Let p(λ) be the characteristic polynomial for A and suppose that A is diagonalizable. Prove that p(A) = 0. (Note: This means that if p(λ) = λ2 + λ + 1
then p(A) = A2 + A + I.) Hint: Show that p(A)x = 0 for all x ∈ Rn .
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Math 317: Linear Algebra
Practice Final Exam
Fall 2015


 
8
1



7. If possible, write the vector x = −4 as a linear combination of u1 = 0,
−3
1
 
 
−1
2



u2 = 4 , and u3 = 1 .
1
−2
8. Prove that any orthogonal set of k vectors is linearly independent.


1 0 0
0
0 2 −2 1 

9. Let A = 
0 2 1 −2. Find the rank and reduced row echelon form of A.
0 1 2
2
10. Determine if the following statements are true or false. If the statement is true,
prove it. If the statement is false, provide a counterexample or explain why the
statement is not true.
(a) If A is invertible, then it’s diagonalizable.
(b) If A is diagonalizable, then it’s invertible.
(c) If A is invertible and diagonalizable, then A−1 is diagonalizable.
(d) If A and B have the same eigenvalues, then they have the same eigenvectors.
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