Civil Engineering Reference
In-Depth Information
If |
A
| is the determinant of the matrix [
A
], then the following equations are
valid when row
k
and column
k
are expanded:
n
n
∑
∑
kj
+
ik
+
A
=−
()
1
a A
and
A
=−
()
1
aA
kj
kj
ik
ik
j
=
1
i
=
1
The basket weave method may be used for a three-by-three determinant
only. Take the sum of the products of the three down-right diagonals minus
the sum of the product of the three up-right diagonals shown as follows:
aaa
aaa
aaa
aaaaa
aaaaa
11
12
13
11
12
13
11
12
=
21
22
23
21
22
23
21
22
aaaaaa
31
32
33
31
32
33
31
32
aaa
aaa
aaa
aaaaa
11
12
13
11
12
13
11
1
2
aaaaa
aaaaa
aa aaaa
=
21
22
23
21
22
23
21
22
31
32
33
31
32
33
31
32
(
)
−
(
)
+
+
aa
a
a aa
+
aa aaaa
+
11 22 33
12
23 31
13
21
32
13
22 31
11 23 32
12
21 33
Also, a determinant may be found by the product of the diagonal of any
triangular matrix.
Example 2.3
Matrix determinants
Find the determinant of the following matrix, [
A
], by expansion of minors
and by the basket weave method.
346
123
212
[]
=
Expansion of row 1 yields:
3
23
12
4
13
22
6
12
A
=
−
+
21
34 3426 61 4
316181
(
)
−−
(
)
+−
(
)
A
A
=−
=+−=
