Question

For each of the following statement, determine it is “always true” or “always false”. Explain your answer.

(a) For any matrix A, the matrices A and −A have the same row space. (b) For any matrix A, the matrices A and −A have the same null space.

(c) For any matrix A, if A has nullity 0, then A is nonsingular.

Answer #1

Indicate whether each statement is True or False.
Briefly justify your answers. Please answer all of questions
briefly
(a) In a vector space, if c⊙⃗u =⃗0, then c= 0.
(b) Suppose that A and B are square matrices and that AB is a
non-zero diagonal matrix. Then A is non-singular.
(c) The set of all 3 × 3 matrices A with zero trace (T r(A) = 0)
is a vector space under the usual matrix operations of addition and
scalar...

Is each statement true or false? If true, explain why; if false,
give a counterexample.
a) A linear system with 5 equations and 4 unknowns is always
inconsistent.
b) If the coefficient matrix of a homogeneous system has a
column of zeroes, then the system has infinitely many solutions.
(Note: a homogeneous system has augmented matrix [A | b] where b =
0.)
c) If the RREF of a homogeneous system has a row of zeroes, then
the system has...

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...

Determine if the following statement is true or false.
Explain your reasoning.
If A if an m x n matrix with linearly independent columns, then
m is greater than or equal to n.

True or False (Please explain)
1. If E and F are elementary matrices then C = E*F is
nonsingular.
2. If A is a 3x3 matrix and
a1+2a2-a3=0 then A must be
singular.
3. If A is a 3x3 matrix and
3a1+a2+4a3=b is consistent.

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...

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...

Discuss the validity of the statement. If the statement is
always true, explain why. If not, give a counterexample. If all
payoffs of a nonstrictly determined matrix game are positive, then
the value of the game is positive.
Is the statement true or false?

True, False or uncertain? Explain whether each of the
following statement is true, false or uncertain. Start your answer
by selecting one of the three statements – “True”, “False” and
“Uncertain” and then provide arguments to justify your selection
(be brief and concise in less than 100 words). You need to make
assumption clear, reasonable and explicit if making any.
a. Nominal interest rates are always higher than real interest
rates. Answer
b. If the lockdown measure due to a further...

(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...

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