Question

Asked Apr 10, 2019

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Explain why G forms a field with respect to addition and multiplication. (You may quote relevant properties of matrix addition and multiplication without proof.)

Why the arithmetic F does not form a field?

Step 1

The two sets are

Step 2

See to the definition of field, a set F is a field that has atleast two elements operated with addition or multiplication such that for each pair of elements a, b belongs to F, there exist unique elements a + b belongs to F and ab belongs to F. For a set F to be a field, each element in F has an additive inverse and a multiplicative inverse in F.

Additive inverse exists for each element in F and G but it is important to see the result for multiplicative inverse.

For C and D in G, CD belongs in G. For multiplicative inverse of D, observe that

Step 3

Here, multiplicative inverse of D is C and simultaneously, multiplicative inverse of C is D.

Therefore, G is a Field as both additive and...

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