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

Here are the answers for each case:

  1. Yes
  2. No
  3. Yes
  4. Yes
  5. Yes
  6. No
  7. Yes
  8. No
  9. No
  10. Yes

Would you like further explanations or details on any of these?

Here are 5 related questions to consider:

  1. What are the key properties required for a set and operation to form a group?
  2. Can matrix multiplication be a group operation for non-square matrices?
  3. How does matrix addition differ from matrix multiplication in terms of group structure?
  4. What is the importance of the determinant in determining group structures for matrices?
  5. Can a set of non-integer real numbers form a group under addition?

Tip: Always check for the identity element and inverses when determining if a structure forms a group.

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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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