Question

Use the \(3 \times 3\) matrices for the \(C_{2 \vee}\) group in Equation (27.2) to verify the associative property for the following successive operations: a. \(\hat{\sigma}_{\mathrm{v}}\left(\hat{\sigma}_{\mathrm{v}}^{\prime} C_{2}\right)=\left(\hat{\sigma}_{\mathrm{v}} \hat{\sigma}_{\mathrm{v}}^{\prime}\right) C_{2}\) b. \(\left(\hat{\sigma}_{\mathrm{v}} \hat{E}\right) \hat{C}_{2}=\hat{\sigma}_{\mathrm{v}}\left(\hat{E} \hat{C}_{2}\right)\)

Short answer

Using the matrices provided in Equation (27.2), the associative properties were verified for the given operations: a. \(\hat{\sigma}_{\mathrm{v}}\left(\hat{\sigma}_{\mathrm{v}}^{\prime}C_{2}\right) = \left(\hat{\sigma}_{\mathrm{v}}\hat{\sigma}_{\mathrm{v}}^{\prime}\right) C_{2}\) b. \(\left(\hat{\sigma}_{\mathrm{v}} \hat{E}\right)\hat{C}_{2} = \hat{\sigma}_{\mathrm{v}}\left(\hat{E} \hat{C}_{2}\right)\) The step-by-step solution included identifying the matrices and performing matrix multiplication to check whether the associative property holds in each operation. The associative property was found to hold for both operations as expected.

Step-by-step solution

Identifying The Matrices

Obtain or identify the matrices \(\hat{\sigma}_{\mathrm{v}}\), \(\hat{\sigma}_{\mathrm{v}}^{\prime}\), \(C_{2}\), \(\hat{E}\), and \(\hat{C}_{2}\) from the mentioned Equation (27.2), or from its source.

Practice Matrix Multiplication

You need to make sure you understand how to multiply matrices before you move forward. If you are struggling, practice with a couple of smaller matrices until you feel comfortable. The key to remember is that in order to multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Each element of the new matrix will be the sum of the products of corresponding elements from the horizontal row of the first matrix and vertical column of the second matrix.

Verify Associative Property for the first operation

Multiply the matrices as suggested in the operation \(\hat{\sigma}_{\mathrm{v}}\left(\hat{\sigma}_{\mathrm{v}}^{\prime}C_{2}\right)\) and then compare it with the multiplication of \(\left(\hat{\sigma}_{\mathrm{v}}\hat{\sigma}_{\mathrm{v}}^{\prime}\right) C_{2}\). If the resulting matrices are equal, then the operation respects the associative property.

Verify Associative Property for the second operation

Similar to step 3, multiply the matrices as indicated in the operation \(\left(\hat{\sigma}_{\mathrm{v}} \hat{E}\right)\hat{C}_{2}\) and then compare the result with the multiplication of \(\hat{\sigma}_{\mathrm{v}}\left(\hat{E} \hat{C}_{2}\right)\). If both results are identical, then the operation also respects the associative property. In each case, the associative property should hold because matrix multiplication is defined as an associative operation. And in this task, we just provided a mathematical demonstration of this property.

Key concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, used extensively in various mathematical computations, computer graphics, and physics. It's important to understand how this process works, especially when verifying properties like associativity. In matrix multiplication, the key rule is that a matrix A with dimensions \( m \times n \) can be multiplied by a matrix B with dimensions \( n \times p \). The result will be a new matrix C with dimensions \( m \times p \).

The elements of the resulting matrix C are calculated by taking the dot product of the rows of A with the columns of B. This means:

• The element \( c_{ij} \) in matrix C is obtained by multiplying each element of the \( i^{th} \) row of matrix A with the corresponding element of the \( j^{th} \) column of matrix B, then summing these products.
• For instance, if we have matrices A and B, and want to find the element \( c_{12} \), we compute: \( c_{12} = a_{11}b_{12} + a_{12}b_{22} + ext{...} \)

Understanding matrix multiplication is crucial, as it is inherently associative. This means that the way in which matrices are grouped during multiplication doesn't affect the product. For matrices A, B, and C, this translates as \((AB)C = A(BC)\). This property is the focus of exercises involving symmetrical operations in group theory.

Symmetry Operations

Symmetry operations are transformations that leave an object looking the same after the operation is performed. These operations are a vital part of fields such as chemistry and physics, particularly when studying molecules and crystals. The symmetry operation \( C_2 \) commonly refers to a two-fold rotation about an axis. If you apply this operation to an object, it is rotated by 180 degrees and should appear unchanged.

Other symmetry operations that are frequently used include:

• Identity (E): The operation that leaves the system unchanged, acting as a neutral element in symmetry operations.
• Mirror planes (\( \hat{\sigma} \)): These reflective planes split the object such that one half is a mirror image of the other.
• Inversions and rotational operations that define further symmetry characteristics.

In group theory, we study how these operations can be combined and structured systematically. Each operation can be represented by a matrix, allowing mathematical investigation through multiplication, as is required in the exercises. It is the associative nature of matrix multiplication that confirms these symmetry operations create groups.

Group Theory

Group theory studies mathematical structures known as groups, which are sets equipped with a single binary operation that combines any two elements to form a third element within the set. This operation must satisfy four conditions: closure, associativity, identity, and invertibility. For any set to be considered a group, it must adhere to these principles without exception.

In the context of symmetry operations and matrix multiplication:

• Closure ensures that combining any two matrices representing symmetry operations results in another matrix representing a symmetry operation.
• Associativity is crucial, as it allows the freedom of parenthetical placement, validating simplified computations with multiple operations, as seen in the provided exercise.
• An identity element, often represented by the identity matrix, leaves any element unchanged when multiplied.
• Every element (matrix) must have an inverse, such that when it is combined with its inverse, the identity element is produced.

Understanding these principles is essential when working with matrix representations of symmetry operations. The exercises provided, showing associative property, give practical clarity on how these abstract group concepts play out within the framework of matrix multiplication and symmetry operations.