Page 475 - Applied Numerical Methods Using MATLAB

464 MATRIX OPERATIONS/PROPERTIES
with
K

c mn = a mk b kn (B.2.2)
k=1

(cf) For this multiplication to be done, the number of columns of A must equal the
number of rows of B.
(cf) Note that the commutative law does not hold for the matrix multiplication, that is,
AB = BA.

B.3 DETERMINANT

The determinant of a K × K (square) matrix A = [a mn ]isdefined by
K K
k+n m+k
det(A) =|A|= a kn (−1) M kn or a mk (−1) M mk (B.3.1)
k=0 k=0
for any fixed 1 ≤ n ≤ K or 1 ≤ m ≤ K

where the minor M kn is the determinant of the (K − 1) × (K − 1) (minor)
matrix formed by removing the kth row and the nth column from A and A kn =
(−1) k+n M kn is called the cofactor of a kn .
In particular, the determinants of a 2 × 2matrix A 2×2 and a 3 × 3matrix
A 3×3 are
2

a k+n
det(A 2×2 ) = 11 a 12 = a kn (−1) M kn = a 11 a 22 − a 12 a 21 (B.3.2)
a 21 a 22

k=1

a
11 a 12 a 13 a a a
det(A 3×3 ) = a 21 a 22 a 23 = a 11 22 a 23 − a 12 21 a 23 + a 13 21 a 22

a 32 a 33 a 31 a 33 a 31 a 32

a 31 a 32 a 33

= a 11 (a 22 a 33 − a 23 a 32 ) − a 12 (a 21 a 33 − a 23 a 31 ) + a 13 (a 21 a 32 − a 22 a 31 )
(B.3.3)
Note the following properties.
ž If the determinant of a matrix is zero, the matrix is singular.
ž The determinant of a matrix equals the product of the eigenvalues of a
matrix.
ž If A is upper/lower triangular having only zeros below/above the diag-
onal in each column, its determinant is the product of the diagonal ele-
ments.
−1
T
ž det(A ) = det(A);det(AB) = det(A)det(B);det(A ) = 1/det(A)