Question

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 echelon form of Q has n leading variables.

(10). If matrix A ∈Rm×n, then rank(A) ≤ max{m,n}.

Answer #1

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...

This is a TRUE-FALSE Question with justification.
If Q is an orthogonal n×n matrix, then Row(Q) =
Col(Q).
The Answer to this is TRUE. I want to know a solid
reasoning/explanation for it.
In one of the answers, it says that " Since Q is orthogonal,
QTQ = I, so Q is invertible, hence Row(Q) = Col(Q) =
Rn. But my question is: Why is it that for an invertible
matrix, Row(Q) = Col (Q) ?
Any other explanation that...

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.

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...

True/ false
a- If the last row in an REF of an augmented matrix is [0 0 0 4
0], then the associated linear system is inconsistent.
b-The equation Ax=b is consistent if the augmented matrix [A b]
has a pivot position in every row.
c-The set Span{v} for a nonzero v is always a line that may or
may not pass through the origin.

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

Let A be an m×n matrix, x a vector in Rn, and b a vector in Rm.
Show that if x1 in Rn is a solution to Ax=b and x2 is a solution to
Ax=⃗0, then x1 +x2 is a solution to Ax=b.

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...

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 =...

Suppose A and B are invertible matrices, with A being m x m and
B being n x n. For any m x n matrix C and any n x m matrix D, show
that :
a) (A + CBD)-1 = A-1-
A-1C(B-1 +
DA-1C)-1DA-1
b) If A, B and A + B are all m x m invertible matrices, then
deduce from (a) above that (A + B)-1 = A-1 -
A-1(B-1 +
A-1)-1A-1

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