Question

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
a_{1}+2a_{2}-a_{3}=0 then A must be
singular.

3. If A is a 3x3 matrix and
3a_{1}+a_{2}+4a_{3}=b is consistent.

Answer #1

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...

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.

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...

In what follows, A and B denote 2 x 2 matrices. Answer each
question below, with justification. No one answer should be more
than a few lines long.
A1. If k is a scalar, how does the determinant of
kA relate to the determinant of A?
A2. Show that the determinant of A + B is not necessarily the
same as det A + det B. (Remark: a single specific counterexample
suffices!)
A3. If A is singular and B is...

Using elementary transformations, determine matrices B and C so
that BAC=I for the matrix A. Use B and C to compute the inverse of
A; that is, take the inverse of both sides of the equation BAC=I
and then solve for A inverse.
I need to find Matrix B, Matrix C, and Inverse of matrix A
A=
1 2 1 1
0 1 2 0
1 2 2 1
0 -1 1 2

Write a Python class Matrix which defines a two-by-two
matrix with float entries as a Python object. The class Matrix has
to be able to display the object on the screen when the print
function is called, and it should include methods determinant(),
trace(), inverse(), characteristic_polynomial(), and
matrix_product(). Furthermore, a user should be able to multiply
the matrix by a constant and be able to add and subtract two
matrices using the usual symbols + and -. Use the following...

a).For the reduction of matrix determine the elementary matrices
corresponding to each operation. M= 1 0 2 1 5
1 1 5 2 7
1 2 8 4 12 b). Calculate the product P of these elementary
matrices and verify that PM is the end result.

write the following matrices as a product of elementary
matrices:
a)
1 2
4 9
b)
1 -2 -1
-1 5 6
5 -4 5
c)
1 0 -2
-3 1 4
2 -3 4

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...

MATLAB:
Do the following with the provided .m file
(a) In the .m file, we have provided three questions. Make sure
to answer them.
(b) Now on the MATLAB prompt, let us create any two 3 × 3
matrices and you can do the following:
X=magic(3);
Y=magic(3);
X*Y
matrixMultiplication3by3(X,Y)
(c) Now write a new function in MATLAB called
matrixMultiplication that can multiply any two n × n matrix. You
can safely assume that we will not test your program with...

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