Question

n x n matrix A, where n >= 3. Select 3 statements from the invertible matrix theorem below and show that all 3 statements are true or false. Make sure to clearly explain and justify your work.

A=

-1 , 7, 9

7 , 7, 10

-3, -6, -4

- The equation A has only the trivial solution.

5. The columns of A form a linearly independent set.

6. The linear transformation x → Ax is one-to-one.

7. The equation Ax = b has at least one solution for each b ∈
R^{n} .

8. The columns of A span R^{n} .

9. The linear transformation maps R^{n} onto
R^{n} .

10. There is an n × n matrix C such that CA = I.

11. There is an n × n matrix D such that AD = I.

12. A^{T} is invertible.

Answer #1

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Let A be an n by n matrix, with real
valued entries. Suppose that A is NOT invertible.
Which of the following statements are true?
?Select ALL correct answers.?
The columns of A are linearly dependent.
The linear transformation given by A is one-to-one.
The columns of A span
Rn.
The linear transformation given by A is onto
Rn.
There is no n by n matrix D such that
AD=In.
None of the above.

et A be an n by n matrix, with real
valued entries. Suppose that A is NOT invertible.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.
The columns of A are linearly independent.The linear
transformation given by A is not one-to-one.The columns of
A span Rn.The linear
transformation given by A is onto
Rn.There is an n by
n matrix D such that
AD=In.None of the above.

Suppose that A is an invertible n by
n matrix, with real valued entries.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.
A is row equivalent to the identity matrix
In.A has fewer than n pivot
positions.The equation
Ax=0 has only the
trivial solution.For some vector b in
Rn, the equation
Ax=b has no
solution.There is an n by n matrix C
such that CA=In.None of the above.

Given that A and B are n × n matrices and T is a linear
transformation. Determine which of the following is FALSE.
(a) If AB is not invertible, then either A or B is not
invertible.
(b) If Au = Av and u and v are 2 distinct vectors, then A is not
invertible.
(c) If A or B is not invertible, then AB is not invertible.
(d) If T is invertible and T(u) = T(v), then u =...

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...

True or False
(5). Suppose the matrix A and B are both invertible, then (A +
B)−1 = A−1 + B−1
. (6). The linear system ATAx = ATb is always consistent for any
A ∈ Rm×n, b ∈Rm .
(7). For any matrix A ∈Rm×n , it satisﬁes dim(Nul(A)) =
n−rank(A).
(8). The two linear systems Ax = 0 and ATAx = 0 have the same
solution set.
(9). Suppose Q ∈Rn×n is an orthogonal matrix, then the row...

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

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.

Consider the matrix A=
−2−2 6]
[−2−3 5]
[3 4−8]
[−7−9 18
(all one matrix)
(a) How many rows ofAcontain a pivot position?
(b) Do the columns ofAspanR4?
(c) Does the equationA ~x=~b have a solution for
every~b∈R^4?
(d) Would the equation A~x=~0 have a nontrivial solution?
(e) Are the columns of A linearly independent?
(~x is vector x)

Prove that for a square n ×n matrix A, Ax = b (1) has one and
only one solution if and only if A is invertible; i.e., that there
exists a matrix n ×n matrix B such that AB = I = B A.
NOTE 01: The statement or theorem is of the form P iff Q, where
P is the statement “Equation (1) has a unique solution” and Q is
the statement “The matrix A is invertible”. This means...

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