Group Structure in Matrix Operations and Sets

Math Problem Statement

Problem 1. Determine whether the binary operation gives a group struc- ture on the given set (just answer "yes" or "no", no reasons needed,)

(1) All 3 x 3 matrices under matrix addition.

(2) All 3 x 3 diagonal matrices under matrix multiplication.

(3) All 3 x 3 diagonal matrices with all diagonals non-zero under matrix multiplication.

(4) All 3 x 3 diagonal matrices with positive diagonals entries under matrix multiplication.

(5) All 5 x 5 upper triangular matrices under matrix addition.

(6) All 5 x 5 upper triangular matrices with positive diagonal entries under matrix multiplication...

(7) All 5 x 5 upper triangular matrices with determinant 1 under matrix multiplication..

(8) All non-negative real numbers under addition.

(9) The set {3" n∈ Z} under multiplication.

(10) The set {3n|n∈Z} under addition.

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Group Theory
Matrix Operations
Binary Operations
Algebraic Structures

Matrix addition
Matrix multiplication
Group axioms: closure, associativity, identity element, inverse element

Group properties
Determinant in matrix groups

Undergraduate Mathematics

Group Theory Analysis of Matrix Addition and Multiplication

Algebraic Structures: Groupoid, Semigroup, Monoid, and Group Classification

Understanding Groupoids, Semigroups, Monoids, and Groups with Integer, Rational, and Real Number Sets

Group Theory Analysis of Matrices and Permutations

Group Theory: Checking Group Properties in a Mathematical System