MAT 442: Test 1 Solutions
1.
Read these instructions carefully. For each type of row operation give the following:
(a) A description of what it does (so we know which row operation you are talking
about)
(b) The determinant of a corresponding elementary matrix
E
(c) State whether or not
E
t
is an elementary matrix
(d) State whether or not
E
is invertible, and how we know that.
Note, you do not have to know the number which goes which each type of row operation,
but you have to cover all types.
Type I operations:
a) Swap row
i
with row
j
with
i
6
=
j
.
b) det(
E
) =

1
c) Yes it is an elementary matrix, in fact,
E
t
=
E
d) Yes it is invertible because we can undo the operation by repeating it:
E
2
=
I
Type II operations:
a) Multiply row
i
by
k
∈
F
where
k
6
= 0
b) det(
E
) =
k
c) Yes it is an elementary matrix, in fact,
E
t
=
E
d) Yes, because we can undo the operation by multiplying the same rwo by 1
/k
,
which exists since
k
6
= 0
Type III operations:
a) Add
k
times row
i
into row
j
where
i
6
=
j
(i.e.,
R
j
←
R
j
+
kR
i
)
b) det(
E
) = 1
c) Yes it is an elementary matrix – the transpose corresponds to adding
k
times row
j
into row
i
d) Yes it is invertible because we can undo the operation by adding

k
times row
i
into row
j
, which corresponds to multiplying by another elementary matrix
2.
Suppose
T
∈ L
(
V, W
)
and
U
∈ L
(
W, Z
)
are linear transformations where
V
,
W
, and
Z
are finitedimensional vector spaces.
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 Fall '09
 Linear Algebra, Algebra, Determinant, Row, Det, Elementary matrix

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