Graphics Reference
In-Depth Information
If
B
is obtained by interchanging two rows or two columns of
A
, then
|
(swapping two rows or columns changes how the hypervolume is swept out and
therefore changes the sign of the determinant).
|
B
| =−|
A
●
If
B
is derived from
A
by adding a multiple of one row (or column) to another
row (or column), then
●
(this addition skews the hypervolume parallel
to one of its faces, thus leaving the volume unchanged).
|
B
| = |
A
|
The determinant of the product of two
n
×
n
matrices
A
and
B
is equal to the
●
product of their respective determinants,
|
AB
| = |
A
||
B
|
.
If
B
is obtained by multiplying a row (or column) of
A
by a constant
k
, then
|
●
B
| =
k
|
A
|
.
If one row of
A
is a multiple of another row, then
|
A
| =
0. The same is true
●
when a column of
A
is a multiple of another column.
For a determinant with a row or column of zeroes,
|
A
| =
0.
●
An effective way of evaluating determinants is to use row and column operations on
the matrix to reduce it to a triangular matrix (where all elements below or above the
main diagonal are zero). The determinant is then the product of the main diagonal
entries. For example, the determinant of
A
,
⎡
⎤
26
250
−
4
−
⎣
⎦
,
A
=
21
−
4
can be evaluated as follows:
4
26
250
−
Adding the second row
to the third row …
det(
A
)
=
−
21
−
4
4
26
250
06
−
1
Adding
2
times the first
row to the second row …
−
=
−
4
4
26
06
−
Adding
1 times the second
row to the third row …
−
=
−
3
06
−
4
