Question

Deside whether the statements below are true or false. If
true, explain why true. If false, give a counterexample.

(a) If a square matrix A has a row of zeros, then A is not
invertible.

(b) If a square matrix A has all 1s down the main diagonal,
then A is invertible.

(c) If A is invertible, then A−1 is invertible.

(d) If AT is invertible, then A is invertible.

Answer #1

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...

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...

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...

1. Determine if the statements are true or false:
a. The eigenvalues of a lower triangular matrix are the diagonal
entries of the matrix.
b. For every square matrix A, the sum of all the eigenvalues of
A is equal to the sum of all the diagonal entries of A.

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...

IDENTIFY EACH OF THESE STATEMENT AS TRUE OR FALSE. If the
statement is true ,explain why .if it is false ,give a
counterexample.(a) if the diagonals of a quatrilateral are
congruent,but only one is the perpendicular of the other,then the
quadrilateral is a kite. (b) if the quadrilateral has exactly one
of reflectional symmetry,then the quadrilateral is a kite. (c) if
the diagonals of a quadrilateral are congruent and bisect each
other,then it is square

Say if it is true or false and explain why
a if a matrix A has row echelon form shown below, then
dim(row(A))= 2
the matrix:
[1 0 0 ]^t [1 1 0 ]^t [-2 8/3 0 ]^t [4 -10/3 0 ]^t

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?

For each of the following short statements, explain whether it
is True or False. If it’s true, explain why. If it’s false, give a
counter-examples or explain why it’s false. (a) (b) (c)
(5 points) Suppose in a game a player has three decision nodes,
with three possible actions at each node: A,B and C. The player has
fewer strategies in a version of the game where C end the game,
than in another version of the game where C...

For each of the following statements, indicate whether it is
true, false, or uncertain and EXPLAIN WHY.
a. In the long-run the typical monopolistically competitive firm
earns no economic profit and that indicates that the firm is
economically (productively) efficient.
b. Monopolists have complete pricing freedom as they seek to
maximize profits.
c. In the short-run, if price drops below the average total
cost, the perfectly competitive firm must shut down
immediately.

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