Page 45 - A Course in Linear Algebra with Applications

1.3: Matrices over Rings and Fields 29

Exercises 1.3
1. Show that the following sets of numbers are fields if the
usual addition and multiplication of arithmetic are used:
(a) the set of all rational numbers;
(b) the set of all numbers of the form a + by/2 where a
and b are rational numbers;
(c) the set of all numbers of the form a + by/^l where
where a and b are rational numbers.

2. Explain why the ring M n (C) is not a field if n > 1.
3. How many n x n matrices are there over the field of two
elements? How many of these are symmetric ? [You will need
the formula l + 2 + 3 + '-- + n = n(n + l)/2; for this see
Example A.l in the Appendix ].
4. Let

/ l 1 1 \ / O i l
A = 0 1 1 and B = 1 1 1
\ 0 1 0 / \ 1 1 0

be matrices over the field of two elements. Compute A + B,
A 2 and AB.

5. Show that the set of all n x n scalar matrices over R with
the usual matrix operations is a field.
6. Show that the set of all non-zero nxn scalar matrices over
R is a group with respect to matrix multiplication.
7. Explain why the set of all non-zero integers with the usual
multiplication is not a group.