12.
There exists a real number
k
such that the matrix
1
[
k
-=-4
k
-=-2
] fails
to
be invertible.
3
13.
There exists a real number
k
such that the matrix
2
[
k
-=-3
k
2]
fails
to
be invertible.
I
1/2].
I
. .
.
14.
A
=
0
1/2
IS
a regu ar transItion matnx.
[
15.
The fonnula det(2A)
=
2 det(
A)
holds for all 2
x
2
matrices
A.
16.
There exists a matrix A such that
17.
Matrix
[3
1
26]
is
invertible.
18.
Matrix
is invertible.
19.
There exists
an
upper triangular 2
x
2 matrix A such
2
that A
=
] .
20.
The function
T
[:]
=
=
is a
linear transformation.
21.
There exists an invertible
n
x
n
matrix with two identi-
cal rows.
22.
If
A
2
=
In,
then matrix A must be invertible.
23.
There exists a matrix A such that A
1
11]
__
[11
2]
2 .
[
24.
There exists a matrix A such that
A
=
] .
25.
The matrix
[I
1]
represents a reflection about a
I
-I
line.
26.
For every regular transition matrix A there exists a tran-
sition matrix
B
such that
AB
=
B.
h
·
[a
b]
[d
-b].
I
27.
T e matnx product
c
d
-c
a
IS
a ways a
scalar multiple
of
h.
28.
There exists a nonzero upper triangular 2 x 2 matrix A
suchthatA
2
=
29.
There exists a positive integer
n
such that
[
0
-I]
n
1
°
=
h
30.
There exists an invertible 2 x 2 matrix A such that
A-I
=
31.
There exists a regular transition matrix A
of
size 3
x
3
such that A
2
=
A.
32.
If
A
is
any transition matrix and
B
is any positive tran-
sition matrix, then
AB
must be a positive transition ma-
trix.
33.
If
matrix
ag
hbe
Jet"]
IS
invertible, then
matrix
[d
a
be]
must
be
invertible as well.
[d
34.
If
A
2
is
invertible, then matrix A itself must be invert-
ible.
35.
If
A
17
=
h,
then matrix A must be
h.
36.
If
A
2
=
h,
then matrix A must be either
h
or
-h.
37.
If
matrix A is invertible, then matrix
SA
must be invert-
ible
as
well.
38.
If
A and
B
are two 4
x
3 matrices such that
Av
=
Bv
for all vectors
v
in
]R3,
then matrices A and
B
must be
equal.
39.
If
matrices A and
B
commute, then the formula A
2
B
=
B
A
2
must hold.
40.
If
A
2
=
A for an invertible
n
x
n
matrix
A,
then A must
be
In.
41.
If
A is any transition matrix such that A
100
is positive,
then A
101
must be positive as well.
42.
If
a transition matrix A is invertible, then A
-I
must be
a transition matrix as well.
43.
If
matrices A and
B
are both invertible, then matrix
A
+
B
must be invertible
as
well.
44.
The equation A
2
=
A holds for all 2 x 2 matrices A
representing a projection.
45.
The equation A
-I
=
A holds for all 2 x 2 matrices A
representing a reflection.
46.
The formula
(Av)·
(Aw)
=
v·
w
holds for all invertible
2 x 2 matrices A and for all vectors
v
and
w
in
]R2.
47.
There exist a 2 x 3 matrix A and a 3 x 2 matrix
B
such
that
AB
=
h
48.
There exist a 3 x 2 matrix A and a 2 x 3 matrix
B
such
that
AB
=
h
49.
If
A
2
+
3A
+413
=
0 for a 3 x 3 matrix
A,
then A must
be invertible.
50.
If
A is an
n
x
n
matrix such that A
2
=
0, then matrix
In
+
A must be invertible.

51.
If
matrix
A
commutes with
B,
and
B
commutes with
C,
then matrix
A
must commute with
C.