Question

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 be an n x n matrix and On be the nxn zero matrix. If d^2 = On, then D = On.

(e) Let A, B, and C be invertible nxn matrices. Then det(ABC) = (det(2B)det(3C^t)) / det(6a^-1)

Answer #1

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.

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...

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...

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.

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...

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

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...

1). Show that if AB = I (where I is the identity matrix) then
A is non-singular and B is non-singular (both A and B are nxn
matrices)
2). Given that det(A) = 3 and det(B) = 2, Evaluate (numerical
answer) each of the following or state that it’s not possible to
determine the value.
a) det(A^2)
b) det(A’) (transpose determinant)
c) det(A+B)
d) det(A^-1) (inverse determinant)

Let A be a 2x2 matrix and suppose that det(A)=3. For each of the
following row operations, determine the value of det(B), where B is
the matrix obtained by applying that row operation to A.
a) Multiply row 1 by -4
b) Add 4 times row 2 to row 1
c) Interchange rows 2 and 1
Resulting values for det(B):
a) det(B) =
b) det(B) =
c) det(B) =

True or False? (Make sure to justify your
answer.)
For this question let ?A be a non-zero ?×?m×n matrix, and ?B an
invertible ?×?n×n matrix.
If {?1,...,??}{x1,...,xk} is a basis for null(?)null(A), then
dim(null(??))=?dim(null(AB))=k.

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