Question

(1 point) Which of the following subsets of {R}^{3x3} are subspaces of {R}^{3x3}?

A. The 3x3 matrices with determinant 0

B. The 3x3 matrices with all zeros in the first row

C. The symmetric 3x3 matrices

D. The 3x3 matrices whose entries are all integers

E. The invertible 3x3 matrices

F. The diagonal 3x3 matrices

Answer #1

Determine if the following subsets are subspaces:
1. The set of differentiable functions such that f´ (0) = 0
2. The set of matrices of size nxn with determinant 0.

Find the dimension of each of the following vector spaces.
a.) The space of all n x n upper triangular matrices A with
zeros in the main diagonal.
b.) The space of all n x n symmetric matrices A.
c.) The space of all n x n matrices A with zeros in the first
and last columns.

Let M be a 3x3 matrix.
(4pts) For which matrix E will the transformation
EM ( from M to EM) perform an
exchange of the first and the third row?
(4pts) What about scaling the second row by a factor
of r ?
(4pts) What about adding a times the first
row to the third row?
(3pts) Can you describe the general forms of these matrices
(that represent elementary row operations)?

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...

(7) Prove the following statements.
(c) If A is invertible and similar to B, then B is invertible
and A−1 is similar to B−1 .
(d) The trace of a square matrix is the sum of the diagonal
entries in A and is denoted by tr A. It can be verified that tr(F
G)=tr(GF) for any two n × n matrices F and G. Prove that if A and B
are similar, then tr A = tr B

please choose your favorite, unique 3x3 Matrix
A containing no more than two 0 entries and having a
nonzero determinant. I suggest choosing a matrix
with integer elements (e.g. not fractions or irrational numbers)
for computational reasons.
What is your matrix A? What is det (A)?
What is AT? What is det (AT)?
Calculate A AT. Show that A AT is
symmetrical.
Calculate AT A
Calculate the determinant of (A AT) and the
determinant of (AT A). Should the determinants...

Select all statements below which are true for all invertible
n×n matrices A and B
A. AB=BA
B. (A+B)^2=A^2+B^2+2AB
C. (In−A)(In+A)=In−A^2
D. 7A is invertible
E. (AB)^−1=A^−1*B^−1
F. A+A^−1 is invertible

1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...

Solve the system
-2x1+4x2+5x3=-22
-4x1+4x2-3x3=-28
4x1-4x2+3x3=30
a)the initial matrix is:
b)First, perform the Row Operation 1/-2R1->R1. The resulting
matrix is:
c)Next perform operations
+4R1+R2->R2
-4R1+R3->R3
The resulting matrix is:
d) Finish simplyfying the augmented mantrix down to reduced row
echelon form. The reduced matrix is:
e) How many solutions does the system have?
f) What are the solutions to the system?
x1 =
x2 =
x3 =

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 =...

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