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136
(viii)
If
A
=
£
12
34
±
, then
A
2
−
5
A
−
2
I
=
0.
Solution.
True.
(ix)
Every
n
×
n
matrix over
R
has a real eigenvalue.
Solution.
False.
(x)
h
217
018
000
i
is diagonalizable.
Solution.
True.
4.47
Let
R
be a commutative ring, let
D
:
Mat
n
(
R
)
→
R
be a determinant
function, and let
A
be an
n
×
n
matrix with rows
α
1
,...,α
n
.D
e
f
ne
d
i
:
R
n
→
R
by
d
i
(β)
=
D
(α
i
i
−
1
,β,α
i
+
1
n
)
.
(i)
If
i
±=
j
and
r
∈
R
, prove that
d
i
(
r
α
j
)
=
0
.
Solution.
Absent.
(ii)
If
i
±=
j
and
r
∈
R
, prove that
d
i
(α
i
+
r
α
j
)
=
D
(
A
)
.
Solution.
Absent.
(iii)
If
r
j
∈
R
, prove that
d
i
(α
i
+
X
j
±=
i
r
j
α
j
)
=
D
(
A
).
Solution.
Absent.
4.48
If
O
is an orthogonal matrix, prove that det
(
O
)
=²
1.
Solution.
Absent.
4.49
If
A
0
is obtained from an
n
×
n
matrix by interchanging two of its rows,
prove that det
(
A
0
)
=−
det
(
A
)
.
Solution.
Absent.
4.50
If
A
is an
n
×
n
matrix over a commutative ring
R
and if
r
∈
R
, prove that
det
(
rA
)
=
r
n
det
(
A
)
. In particular, det
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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