Page 133 - Numerical Analysis Using MATLAB and Excel

Minors and Cofactors

Example 4.8

Compute the determinant of using the elements of the first row.
A

1 2 – 3
A = 2 – 4 2 (4.25)
– 1 2 – 6
Solution:

)
detA = 1 – 4 2 – 2 2 2 – 3 2 – 4 = 1 × 20 – 2 × – ( 10 – 3 × 0 = 40
–
2 – 6 – 1 6 – 1 2
Check with MATLAB:

A=[1 2 −3; 2 −4 2; −1 2 −6]; det(A) % Define matrix A and compute detA

ans =
40
The MATLAB user−defined function file below can be used to compute the determinant of a
4 × 4 matrix.

We must use the above procedure to find the determinant of a matrix of order 4 or higher.
A
Thus, a fourth−order determinant can first be expressed as the sum of the products of the ele-
ments of its first row by its cofactor as shown below.

a 11 a 12 a 13 a 14 a a a a a a
a a a a 22 23 24 12 13 14
A = 21 22 23 24 = a 11 a a a a – 21 a a a (4.26)
a 31 a 32 a 33 a 34 a 32 a 33 a 34 a 32 a 33 a 34
a 41 a 42 a 43 a 44 42 43 44 42 43 44

a 12 a 13 a 14 a 12 a 13 a 14
+a 31 a 22 a 23 a 24 – a 41 a 22 a 23 a 24

a 42 a 43 a 44 a 32 a 33 a 34

Determinants of order five or higher can be evaluated similarly.

Example 4.9

Compute the value of the determinant

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−15
Copyright © Orchard Publications