This set of Discrete Mathematics Puzzles focuses on “Groups – Existence of Identity & Inverse”.

1. In a group there must be only __________ element.

a) 1

b) 2

c) 3

d) 5

View Answer

Explanation: There can be only one identity element in a group and each element in a group has exactly one inverse element. Hence, two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements.

2. _____ is the multiplicative identity of natural numbers.

a) 0

b) -1

c) 1

d) 2

View Answer

Explanation: 1 is the multiplicative identity of natural numbers as a⋅1=a=1⋅a ∀a∈N. Thus, 1 is the identity of multiplication for the set of integers(Z), set of rational numbers(Q), and set of real numbers(R).

3. An identity element of a group has ______ element.

a) associative

b) commutative

c) inverse

d) homomorphic

View Answer

Explanation: By the definition of all elements of a group have an inverse. For an element, a in a group G, an inverse of a is an element b such that ab=e, where e is the identity in the group. The inverse of an element is unique and usually denoted as -a.

4. __________ matrices do not have multiplicative inverses.

a) non-singular

b) singular

c) triangular

d) inverse

View Answer

Explanation: The rational numbers are an extension of the integer numbers in which each non-zero number has an inverse under multiplication. A 3 × 3 matrix may or may not have an inverse under matrix multiplication. The matrices which do not have multiplicative inverses are termed as singular matrices.

5. If X is an idempotent nonsingular matrix, then X must be ___________

a) singular matrix

b) identity matrix

c) idempotent matrix

d) nonsingular matrix

View Answer

Explanation: Since X is idempotent, we have X

^{2}=X. As X is nonsingular, it is invertible. Thus, the inverse matrix X

^{-1}exists. Then we have, I=X

^{-1}X = X

^{-1}X2=IX=X.

6. If A, B, and C are invertible matrices, the expression (AB^{-1})^{-1}(CA^{-1})^{-1}C2 evaluates to ____________

a) BC

b) C^{-1}BC

c) AB^{-1}

d) C^{-1}B

View Answer

Explanation: Using the properties (AB)

^{-1}=b

^{-1}A

^{-1}and (A

^{-1})

^{-1}=A, we may have,

(AB

^{-1})

^{-1}(CA

^{-1})

^{-1}C2

=(B

^{-1})

^{-1}A

^{-1}(A

^{-1})

^{-1}C

^{-1}C2

=BA

^{-1}AC

^{-1}C2

=BIC=BC [As, A

^{-1}A=I].

7. If the sum of elements in each row of an n×n matrix Z is zero, then the matrix is ______________

a) inverse

b) non-singular

c) additive inverse

d) singular

View Answer

Explanation: By the definition, an n×n matrix A is said to be singular if there exists a nonzero vector v such that Av=0. Otherwise, it is known that A is a nonsingular matrix.

8. ___________ are the symmetry groups used in the Standard model.

a) lie groups

b) subgroups

c) cyclic groups

d) poincare groups

View Answer

Explanation: A symmetry group can encode symmetry features of a geometrical object. The group consists of the set of transformations that leave the object unchanged. Lie groups are such symmetry groups used in the standard model of particle physics.

9. A semigroup S under binary operation * that has an identity is called __________

a) multiplicative identity

b) monoid

c) subgroup

d) homomorphism

View Answer

Explanation: Let P(S) is a commutative semigroup has the identity e, since e*A=A=A*e for any element A belongs to P(S). Hence, P(S) is a monoid.

10. An element a in a monoid is called an idempotent if ______________

a) a^{-1}=a*a^{-1}

b) a*a^{2}=a

c) a^{2}=a*a=a

d) a^{3}=a*a

View Answer

Explanation: An algebraic structure with a single associative binary operation and an Identity element are termed as a monoid. It is studied in semigroup theory. An element x in a monoid is called idempotent if a

^{2}= a*a = a.

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