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04.04.1
Chapter 04.04
Unary Matrix Operations
After reading this chapter, you should be able to:
1.
know what unary operations means,
2.
find the transpose of a square matrix and it’s relationship to symmetric matrices,
3.
find the trace of a matrix, and
4.
find the determinant of a matrix by the cofactor method.
What is the transpose of a matrix?
Let
]
[
A
be a
n
m
×
matrix.
Then
]
[
B
is the transpose of the
]
[
A
if
ij
ij
a
b
=
for all
i
and
j
.
That is, the
th
i
row and the
th
j
column element of
]
[
A
is the
th
j
row and
th
i
column
element of
]
[
B
.
Note,
]
[
B
would be a
m
n
×
matrix.
The transpose of
]
[
A
is denoted by
T
]
[
A
.
Example 1
Find the transpose of
=
27
7
16
6
25
15
10
5
2
3
20
25
]
[
A
Solution
The transpose of
]
[
A
is
[ ]
=
27
25
2
7
15
3
16
10
20
6
5
25
T
A
Note, the transpose of a row vector is a column vector and the transpose of a column vector
is a row vector.
Also, note that the transpose of a transpose of a matrix is the matrix itself, that is,
[ ]
( ) [ ]
A
A
=
T
T
.
Also,
( ) ( )
T
T
T
T
T
;
cA
cA
B
A
B
A
=
+
=
+
.

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