Page 86 - A Course in Linear Algebra with Applications

70 Chapter Three: Determinants

8. If A is the n x n matrix

/ 0 0 • • • 0 a x \
0 0 ••• a 2 0

\ a n 0 ••• 0 0 /

n n 1 2
(
show that det(A) = -l) ( - )/ aia 2 • • • a n .
9. Write down the permutation matrix that represents the
permutation 3, 1, 4, 5, 2.
10. Let ii,..., i n be a permutation of 1,..., n , and let P be
the corresponding permutation matrix. Show that for any n x
n matrix A the matrix AP is obtained from A by rearranging
the columns according to the scheme Cj —> Cj..

11. Prove that the sign of a permutation equals the determi-
nant of the corresponding permutation matrix.

12. Prove that every permutation matrix is expressible as a
product of elementary matrices of the type that represent row
or column interchanges.
1 T
13. If P is any permutation matrix, show that P" = P .
[Hint: apply Exercise 10].

3.2 Basic Properties of Determinants

We now proceed to develop the theory of determinants,
establishing a number of properties which will allow us to
compute determinants more efficiently.

Theorem 3.2.1
If A is an n x n matrix, then

T
&et(A ) =det(A).