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AMS210.01.
Practice Midterm
Andrei Antonenko
1. A square matrix
Q
is called orthogonal if
QQ
>
=
Q
>
Q
=
I
.
(a) Give an example of an orthogonal matrix.
Solution:
Identity matrix is orthogonal: since
I
>
=
I
, we have
II
>
=
I
>
I
=
II
=
I
. Of
course, you can ﬁnd other orthogonal matrices.
(b) A matrix is called
inversion matrix
if it is obtained from the identity matrix by inter
changing two of its rows. Prove that any inversion matrix is orthogonal.
Solution:
Let
E
be an inversion matrix, obtained from
I
by interchanging rows, say,
i
and
j
. Since interchanging rows of
I
is the same as interchanging columns of
I
,
E
>
=
E
. Also,
to get
I
from
E
we have to interchange the same rows –
i
and
j
, and thus
E

1
=
E
, and
thus
EE
>
=
EE
=
EE

1
=
I
.
(c) Prove that the product of two orthogonal matrices is an orthogonal matrix.
Solution:
Let
A
and
B
be orthogonal matrices. It means that
AA
>
=
I
and
BB
>
=
I
.
Then (
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 Spring '03
 Andant

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