(a) M = Q; a °b = a + b - 3ab (b) M = { A ∈ R^2x2 | det A = 1}; A °B = AB (usual matrix multiplication).

Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter12: Algebra Of Matrices
Section12.3: M X N Matrices
Problem 49PS
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Each of the following pairs is composed of a non-empty set M and some binary operation ° on this set. In which case does the set M not contain a neutral element for the operation °?

(a) M = Q; a °b = a + b - 3ab

(b) M = { A ∈ R^2x2 | det A = 1}; A °B = AB (usual matrix multiplication).

(c) M = R[x]; p°q = p △q (polynomial composition).

(d) M = R[x]; p °q = pq (usual polynomial multiplication).

ž(e) M = {(a,b) ∈ R^2 | a + b = 1 }; (a,b) °(c,d) = (a+c,b+d) (usual addition of ordered pairs).

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