SUMMARY
The order of the subgroup generated by the 4x4 matrix A, defined as | 0 1 0 0 |, | 0 0 0 1 |, | 0 0 1 0 |, | 1 0 0 0 |, is determined to be 3. This conclusion is reached by calculating A^3, which results in the identity matrix e, defined as | 1 0 0 0 |, | 0 1 0 0 |, | 0 0 1 0 |, | 0 0 0 1 |. The process involves multiplying the matrix A by itself three times to achieve the identity, confirming that the order of the subgroup is indeed 3.
PREREQUISITES
- Understanding of matrix multiplication
- Familiarity with group theory concepts, specifically the order of a group element
- Knowledge of 4x4 matrices and their properties
- Ability to compute powers of matrices
NEXT STEPS
- Study the concept of the order of a group element in abstract algebra
- Learn matrix multiplication techniques, particularly for larger matrices
- Explore the properties of the multiplicative group of matrices
- Investigate other examples of subgroup orders in linear algebra
USEFUL FOR
Students and educators in linear algebra, mathematicians focusing on group theory, and anyone interested in the properties of matrices and their applications in abstract algebra.