Page 40 - A Course in Linear Algebra with Applications

24 Chapter One: Matrix Algebra

16. Show that a n n x n matrix A which commutes with every
other n x n matrix must be scalar. [Hint: A commutes with
the matrix whose (i,j) entry is 1 and whose other entries are
all 0.]

17. (Negative powers of matrices) Let A be an invertible ma-
l n
trix. If n > 0, define the power A~ n to be (A~ ) . Prove that
1
A-n = (A*)' .
18. For each of the following matrices find the inverse or show
that the matrix is not invertible:

«G9= <21)-

19. Generalize the laws of exponents to negative powers of an
invertible matrix [see Exercise 2.]
20. Let A be an invertible matrix. Prove that A T is invertible
T l 1 T
and (A )~ = {A- ) .
3
21. Give an example of a 3 x 3 matrix A such that A = 0,
but A 2 ^ 0.

1.3 Matrices over Rings and Fields

Up to this point we have assumed that all our matrices
have as their entries real or complex numbers. Now there are
circumstances under which this assumption is too restrictive;
for example, one might wish to deal only with matrices whose
entries are integers. So it is desirable to develop a theory
of matrices whose entries belong to certain abstract algebraic
systems. If we review all the definitions given so far, it be-
comes clear that what we really require of the entries of a
matrix is that they belong to a "system" in which we can add
and multiply, subject of course to reasonable rules. By this we
mean rules of such a nature that the laws of matrix algebra
listed in Theorem 1.2.1 will hold true.