Problem 1.[10pts] The term ”matrix” is always assumed to have all entries being real numbers. Determine whether the binary operation ∗ gives a group structure on the given set (just answer ”yes” or ”no”, no reasons needed,) (1) All 3 × 3 matrices under matrix addition.

(2) All 3 × 3 diagonal matrices under matrix multiplication.
(3) All 3 × 3 diagonal matrices with all diagonals non-zero under matrix multiplication.
(4) All 3 × 3 diagonal matrices with negative diagonals entries under matrix multiplication.
(5) All 5 × 5 upper triangular matrices under matrix addition.
(6) All 5 × 5 upper triangular matrices with positive diagonal entries under matrix multiplication.
(7) All 5 × 5 upper triangular matrices with determinant 1 under matrix multiplication.
(8) All negative real numbers under addition.
(9) The set {π^n | n ∈ Z} under multiplication.
(10) The set {7^n | n ∈ Z} under addition.

Here are the answers for each part:

1. **No** (All 3×3 matrices under matrix addition)
2. **No** (All 3×3 diagonal matrices under matrix multiplication)
3. **Yes** (All 3×3 diagonal matrices with all diagonals non-zero under matrix multiplication)
4. **Yes** (All 3×3 diagonal matrices with negative diagonal entries under matrix multiplication)
5. **Yes** (All 5×5 upper triangular matrices under matrix addition)
6. **No** (All 5×5 upper triangular matrices with positive diagonal entries under matrix multiplication)
7. **Yes** (All 5×5 upper triangular matrices with determinant 1 under matrix multiplication)
8. **No** (All negative real numbers under addition)
9. **Yes** (The set {π^n | n ∈ Z} under multiplication)
10. **No** (The set {7^n | n ∈ Z} under addition)

Do you want further details or have any questions?

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**Related Questions:**

1. What are the group axioms needed to confirm a group structure?
2. Why does matrix addition on 3x3 matrices fail to form a group?
3. How does determinant influence the group structure in matrix multiplication?
4. What properties make diagonal matrices with non-zero diagonals a group under multiplication?
5. How does the closure property apply to upper triangular matrices under matrix addition?

**Tip:** In matrix operations, always check for identity and inverse elements to determine if a set forms a group under a binary operation.

Problem 1.[10pts] The term ”matrix” is always assumed to have all entries being real numbers. Determine whether the binary operation ∗ gives a group structure on the given set (just answer ”yes” or ”no”, no reasons needed,) (1) All 3 × 3 matrices under matrix addition. (2) All 3 × 3 diagonal matrices under matrix multiplication. (3) All 3 × 3 diagonal matrices with all diagonals non-zero under matrix multiplication. (4) All 3 × 3 diagonal matrices with negative diagonals entries under matrix multiplication. (5) All 5 × 5 upper triangular matrices under matrix addition. (6) All 5 × 5 upper triangular matrices with positive diagonal entries under matrix multiplication. (7) All 5 × 5 upper triangular matrices with determinant 1 under matrix multiplication. (8) All negative real numbers under addition. (9) The set {π^n | n ∈ Z} under multiplication. (10) The set {7^n | n ∈ Z} under addition.

Analyze whether various sets of matrices, including 3x3 and 5x5 matrices, form a group under matrix addition or multiplication. This problem explores group theory concepts such as closure and identity, focusing on sets like diagonal matrices and upper triangular matrices.

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