|A|= (1)(−5) + (0)(5) + (1)(−5) = − 10
|A|= (−1)(2) + (1)(−4) + (2)(−2) = − 10
|A|= (3)(−1) + (2)(−3) + (−1)(1) = − 10
and using a column expansion there results
|A|= (1)(−5) + (−1)(2) + (3)(−1) = − 10
|A|= (0)(5) + (1)(−4) + (2)(−3) = − 10
|A|= (1)(−5) + (2)(−2) + (−1)(1) = − 10.
Observe also that if the elements from any row (or column) are multiplied by
the cofactors from a different row (or column), then the sum of these elements is
zero. For example, row 1 multiplied by the cofactors from row 2 gives
(1)(2) + (0)(−4) + (1)(−2) = 0.
Another example is row 2 multiplied by the cofactors from row 3
(−1)(−1) + (1)(−3) + (2)(1) = 0.
These results may be further illustrated by calculating the matrix product
AC T=
|A| 0 0
0 |A| 0
0 0 |A|
=|A|diag (1 , 1 ,1) = |A|I (10 .20)
Example 10-20.
Find the determinant of the matrix
A=
1 0 0 2 6
1 1 0 3 9
−2 0 1 − 3 − 9
0 −1 0 2 1
1 0 1 4 12
.
Solution: Utilizing property 4, one can multiply any row by a constant and add the
result to any other row without changing the value of the determinant. Perform the
following operations on the above determinant: (a) subtract row 1 from row 5 (b)
