Question

Decide if each of the following statements are true or false. If a statement is true, explain why it is true. If the statement is false, give an example showing that it is false.

(a) Let A be an n x n matrix. One root of its characteristic polynomial is 4. The dimension of the eigenspace corresponding to the eigenvalue 4 is at least 1.

(b) Let A be an n x n matrix. A is not invertible if and only if 0 is one of its eigenvalues (Hint: Review the Invertible Matrix Theorem)

(c) A 2 x 2 matrix must have two distinct eigenvalues.

Answer #1

The questions this week is about diagonalizability of matrices
when we have fewer than n different eigenvalues. Recall the
following facts as starting points:
• An n × n matrix is diagonalizable if and only if it has n
linearly independent eigenvectors.
• Eigenvectors with different eigenvalues must be linearly
independent.
• The number of times an eigenvalue appears as a root of the
characteristic polynomial is at least the dimension of the
corresponding eigenspace, and the total degree of...

(6) Label each of the following statements as True or
False. Provide justification
for your response.
(b) True/False The scalar λ is an eigenvalue of a
square matrix A if and
only if the equation (A − λIn)x = 0 has a nontrivial
solution.
(c) True/False If λ is an eigenvalue of a matrix A, then there is
only
one nonzero vector v with Av = λv.
(d) True/False The eigenspace of an eigenvalue of an n × n matrix...

6. Let A = 3 −12 4 −1 0 −2 −1 5 −1 . 1 (a) Find all
eigenvalues of A5 (Note: If λ is an eigenvalue of A, then λ n is an
eigenvalue of A n for any integer n.). (b) Determine whether A is
invertible (Check if the constant term of the characteristic
polynomial χA(λ) is non-zero.). (c) If A is invertible, find (i)
A−1 using the Cayley-Hamilton theorem (ii) All the eigenvalues...

True or false; for each of the statements below, state whether
they are true or false. If false, give an explanation or example
that illustrates why it's false.
(a) The matrix A = [1 0] is not invertible.
[1 -2]
(b) Let B be a matrix. The rowspaces row (B), row (REF(B)) and
row (RREF(B)) are all equivalent.
(c) Let C be a 5 x 7 matrix with nullity 3. The rank of C is
2.
(d) Let D...

2. For each
3*3 matrix and each eigenvalue below construct a basis for the
eigenspace Eλ.
A= (9 42 -30 -4 -25 20
-4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2
3. Construct a 2×2
matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and
−3, respectively.
4. Let A be the 6*6
diagonal matrix below. For each eigenvalue, compute the
multiplicity of λ as a root of the...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Which of the following are NECESSARY CONDITIONS for an n x n
matrix A to be diagonalizable?
i) A has n distinct eigenvalues
ii) A has n linearly independent eigenvectors
iii) The algebraic multiplicity of each eigenvalue equals its
geometric multiplicity
iv) A is invertible
v) The columns of A are linearly independent
NOTE: The answer is more than 1 option.

Determine if each of the following statements is true or false.
If a statement is true, then write a formal proof of that
statement, and if it is false, then provide a counterexample that
shows its false.
1) For each integer a there exists an integer
n such that a divides (8n +7) and
a divides (4n+1), then a divides 5.
2)For each integer n if n is odd, then 8
divides (n4+4n2+11).

For each of the following statements decide whether they are
True or False and give a short argument if True, or counter example
if False.
(1) ∀n ∈ Z, ∃m ∈ Z, n + m ≡ 1 mod 2.
(2) ∀n ∈ Z, ∃m ∈ Z, (2n + 1)^2 = 2m − 1.
(3) ∃n ∈ Z, ∀m > n, m^2 > 100m.

5. Determine whether the following statements are TRUE or FALSE.
If the statement is TRUE, then explain your reasoning. If the
statement is FALSE, then provide a counter-example. a) The
amplitude of f(x)=−2cos(X- π/2) is -2 b) The period of
g(x)=3tan(π/4 – 3x/4) is 4π/3.
. c) If limx→a f (x) does not
exist, and limx→a g(x) does not exist, then limx→a (f (x) + g(x))
does not exist. Hint: Perhaps consider the case where f and g are
piece-wise...

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