Comparing corresponding entries in the first column, we obtain
a
+
b
√
3 =
a
−
c
√
3 and
c
+
d
√
3 =
a
√
3 +
c
, which gives
b
=
−
c
and
d
=
a
. In that case entries in the second
column are automatically equal. We conclude that
B
has the form
B
=
a
−
c
c
a
for arbitrary numbers
a
and
c
.
As in class we conclude that this matrix represents the composition of a rotation and
a dilation. To see this, it is enough to take
r
=
√
a
2
+
c
2
, and find an angle
θ
so that
a
=
r
cos
θ
,
c
=
r
sin
θ
. Then we have:
B
=
r
0
0
r
cos
θ
−
sin
θ
sin
θ
cos
θ
.
4. We perform the algorithm given in class:
⎛
⎝
1
1
1

1
0
0
3
2
0

0
1
0
0
0
1

0
0
1
⎞
⎠
subtract 3 times row I from row II
⎛
⎝
1
1
1

1
0
0
0
−
1
−
3

−
3
1
0
0
0
1

0
0
1
⎞
⎠
multiply row II by
−
1
⎛
⎝
1
1
1

1
0
0
0
1
3

3
−
1
0
0
0
1

0
0
1
⎞
⎠
subtract row II from row I
⎛
⎝
1
0
−
2

−
2
1
0
0
1
3

3
−
1
0
0
0
1

0
0
1
⎞
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 Fall '08
 lee
 times row

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