Game Development Reference
In-Depth Information
6.1.2
Minors and Cofactors
Before we can look at determinants in the general case, we need to introduce
some other constructs: minors and cofactors.
Assume M is a matrix with r rows and c columns. Consider the matrix
obtained by deleting row i and column j from M . This matrix will obviously
have r − 1 rows and c − 1 columns. The determinant of this submatrix,
denoted M {ij} is known as a minor of M . For example, the minor M {12}
is the determinant of the 2 × 2 matrix that is the result of deleting row 1
and column 2 from the 3 × 3 matrix M :
2
4 −4
3
−3
3
A minor of a 3 × 3
matrix
0
−2
5
{12}
M =
0
2
−2
=⇒
M
=
= 2.
1
−1
1
4
−1
The cofactor of a square matrix M at a given row and column is the
same as the corresponding minor, but with alternating minors negated:
{ij}
= (−1) i+j M
{ij}
C
.
(6.4)
Matrix cofactor
{ij}
As shown in Equation (6.4), we use the notation C
to denote the cofactor
of M in row i, column j. The (−1) (i+j) term has the effect of negating every
other cofactor in a checkerboard pattern:
2
4
3
5 .
+
+
+
+
+
+
+
+
.
.
.
.
. . .
In the next section, we use minors and cofactors to compute determi-
nants of an arbitrary dimension n×n, and again in Section 6.2 to compute
the inverse of a matrix.
6.1.3
Determinants of Arbitrary N × N Matrices
Several equivalent definitions exist for the determinant of a matrix of arbi-
trary dimension n × n. The definition we consider here expresses a deter-
minant in terms of its cofactors. This definition is recursive, since cofactors
are themselves signed determinants. First, we arbitrarily select a row or
column from the matrix. Now, for each element in the row or column,
we multiply this element by the corresponding cofactor. Summing these
products yields the determinant of the matrix. For example, arbitrarily
 
Search WWH ::




Custom Search