LINEAR ALGEBRA EXAM 4 version 1 name:
1. Suppose A is a square real matrix and A
−46
6
= 6
−46
6
Select the true statement/s
o
o
A A13
−46
6
= (6)13
−46
6
B A7
−46
6
= 6
−46
6
7
C A7
−46
6
= (6)7
−46
6
D A6
−46
6
= (−7)13
−46
6
E A13
−46
6
= (−7)6
−46
6
F A6
−46
6
= (6)13
−46
6
G none of these
2. Suppose B is the set of COLUMNS of matrix M = −1 1 −10 6 8
0 0 0
and V be the space spanned by linear combi-
nations ofB using real number scalars. Select true statement/s
o
o
A
−30
5
is in spanned by columns of M
B
−31
5
is spanned by columns of M
C
−34
−1
is spanned by columns of M
D
−34
−3
is not spanned by columns of M
E rank(M) = 2
F
4−4
4
is spanned by columns of M
G rank(M) = 4
H
10
0
IS spanned by columns of M
I None of These
3. Suppose D =
12
1
,
−20
0
,
0−1
0
and B =
10
0
,
01
0
,
00
1
Suppose
v⃗B =
02
−2
Determine the coordinates of this vector subject to basis D.
o
o
A v⃗D =
−2−1
−6
B v⃗D =
−6−2
−11
C v⃗D =
02
−2
D None of These
4. Suppose B =
0−3
4
,
−110
−1
and V be the space
spanned by linear combinations of B using real numbe
scalars. In other words
V =
α
0−3
4
+ β
−110
−1
: α, β ∈ R
with the standard addition and the standard scalar multipli-
cation. Select true statement/s
o
o
A
05
−15
∈ V
B
0−2
1
∈ V
C
−641
16
∈ V
D
−859
20
∈ V
E
−892
−24
∈ V
F
−763
4
∈ V
G V is a vector space over R
H None of These
5.
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LINEAR ALGEBRA EXAM 4 version 1 (page 2/ ??)
Suppose P and Q are a real 2 × 2 matrices. Suppose I is the
2x2 identity matrix, and Suppose
PQ = I
Select all necessarily true statements.
o
o
A P +Q = 0
B (PQ)3 = P 3Q3
C P is invertible
D (PQ)2 = PQ
E P−1 does not exist
F (PQ)2 = P 2Q2
G (don’t even try this if you have a small soul)
QP = I
H det(P ) > 0
I None of These
6. Suppose A is a square matrix, and det(A) ̸= 0 then
A−1 is given by
o
o
A
A−1 = det(A) ·Adj(A)
B
A−1 =
1
det(A)
Adj(A)
C none of these
7. Suppose linear transformation T : R2 → R2 with T
[
x
y
]
=[
7x+ y
4x+ y
]
Assume A is the matrix that represents the tran-
formation T so that for ever vector v⃗ we have T (v⃗) = Av⃗ Select
the true statement/s
o
o
A T
[
0
1
]
=
[
1
1
]
B T
[
0
1
]
=
[
−1
3
]
C A =
[
1 −1
0 3
]
D A =
[
1 1
−2 1
]
E T
[
1
0
]
=
[
1
0
]
F T
[
1
0
]
=
[
7
4
]
G A =
[
7 1
4 1
]
H None of These
8. Suppose
A =
[
0 6
−1 −1
]
If it exists, find the inverse, A−1
o
o
A
[
−16 −1
1
6 0
]
B inverse does not exist
C
[
1
6 −
1
12
1 0
]
D
[
1
44 −
7
44
3
22
1
22
]
E none of these
9. Suppose V =
x1
x2
x3
x4
: x1 = x3 + x4, x2 = x3 − x4
. oo
A A basis for V is
−1
0
1
1
,
−1
1
0
2
B A basis for V is
1
1
1
0
,
1
−1
0
1
C A basis for R4 is Basis(V ) ∪Basis(V ⊥)
D V is a subspace of R4
E A basis for R4 is
1
1
1
0
,
1
−1
0
1
,
−1
0
1
1
,
−1
1
0
2
F A basis for V ⊥ is
−1
0
1
1
,
−1
1
0
2
G none of these
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