Page 85 - A Course in Linear Algebra with Applications

3.1: The Definition of a Determinant 69

Exercises 3.1
1. Is the permutation 1, 3, 8, 5, 2, 6, 4, 7 even or odd? What
is the corresponding term in the expansion of det^a^^s)?
2. The same questions for the permutation 8, 5, 3, 2, 1, 7, 6,
9,4.
3. Use the definition of a determinant to compute

1 - 3 0
2 1 4
-1 0 1

4. How many additions, subtractions and multiplications are
needed to compute a n n x n determinant by using the defini-
tion?
5. For the matrix

2 3
4 3
-1 2

find the minors Mi 3 , M 23 and M 33 , and the corresponding
cofactors Ai 3, A 23 and A 33.

6. Use the cofactors found in Exercise 5 to compute the de-
terminant of the matrix in that problem.

7. Use row or column expansion to compute the following
determinants:

1 - 2 3 4
- 2 2 3
(a) 2 2 1 , (b) 2 0 1 2 3
1
0 - 1
0 0 5
0 1 1 3
1 - 2 3 4
0 4 2 1
(c) 0 0 - 3 1
0 0 0 3