Page 46 - Numerical Methods for Chemical Engineering

The determinant 35

Property IV
If two rows (columns) of A are swapped to form B, det(B) =−det(A).

Property V
If two rows (columns) of A are the same, then det(A) = 0.

Property VI
If we decompose a (m) , the row of a matrix A,as
a (m) = b (m) + d (m) (1.174)

to form the matrices
a a a
 (1)   (1)   (1) 
. . .
. . .
     
 .   .   . 
 (m)   (m)   (m) 
A =  a  B =  b  D =  d 
     
. . .
. . .
     
 .   .   . 
a (N) a (N) a (N)
(1.175)
then
det(A) = det(B) + det(D) (1.176)
Property VII

If a matrix B is obtained from A by adding c times one row (column) of A to another row
(column) of A, then det(B) = det(A). That is, elementary row operations do not change the
value of the determinant.

Property VIII
det(AB) = det(A) × det(B) (1.177)

Property IX

If A is upper triangular or lower triangular, det(A) equals the product of the elements on the
principal diagonal, det(A) = a 11 × a 22 ×· · · × a NN .

Computing the determinant value

Properties VI–IX give us the fastest method to compute the determinant. Note that the gen-
eral formula for det(A) is a sum of N! nonzero terms, each requiring N scalar multiplications,
and is therefore very costly to evaluate. Since Gaussian elimination merely consists of a
sequence of elementary row operations that by property VII do not change the determinant