Page 132 - Numerical Analysis Using MATLAB and Excel

Chapter 4 Matrices and Determinants

α , 21 α , 22 α , 23 α , 31 α , 32 and α 33
are defined similarly.

Example 4.7
Given that
–
1 2 3
A = 2 – 4 2 (4.19)
– 1 26
–
compute its cofactors.

Solution:
α 11 – ( = 1 ) 1 + 1 – 4 2 = 20 α 12 – ( = 1 ) 1 + 2 22 = 10 (4.20)
–
–
26 – 1 6
–
α 13 – (= 1 ) 1 + 3 24 = 0 α 21 – ( = 1 ) 2 + 1 23– = 6 (4.21)
– 1 2 26
–
–
α 22 – ( = 1 ) 2 + 2 13 = – 9 α 23 – ( = 1 ) 2 + 3 12 = – 4 (4.22)
–
– 1 6 – 1 2

–
α 31 – ( = 1 ) 3 + 1 23 = – 8 α 32 – ( = 1 ) 3 + 2 13– = – 8 (4.23)
,
– 4 2 22

α 33 – (= 1 ) 3 + 3 12 = – 8 (4.24)
–
24
It is useful to remember that the signs of the cofactors follow the pattern

+ − + − +
− + − + −
+ − + − +
− + − + −
+ − + − +

that is, the cofactors on the diagonals have the same sign as their minors.

Let be a square matrix of any size; the value of the determinant of is the sum of the products
A
A
obtained by multiplying each element of any row or any column by its cofactor.

4−14 Numerical Analysis Using MATLAB® and Excel®, Third Edition

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