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.
Solution
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Math Problem Analysis
Mathematical Concepts
Group Theory
Matrix Operations
Binary Operations
Algebraic Structures
Formulas
Matrix addition
Matrix multiplication
Group axioms: closure, associativity, identity element, inverse element
Theorems
Group properties
Determinant in matrix groups
Suitable Grade Level
Undergraduate Mathematics
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