Problem 10 - Algebra: Pure and Applied

Question

Show that \(G L(2, Q)\) is non-Abelian.

Step-by-Step Solution

Answer

GL(2, Q) is non-Abelian because some matrices do not commute.

Step 1: Identify the Groups Involved

First, we identify that we are dealing with the general linear group of 2x2 matrices, denoted as \(GL(2, \mathbb{Q})\), where \(\mathbb{Q}\) represents the set of rational numbers. This group consists of all 2x2 invertible matrices with rational entries.

Step 2: Define Properties of an Abelian Group

Recall that for a group to be Abelian, or commutative, for every pair of elements \(a\) and \(b\) in the group, the equation \(ab = ba\) should hold. If we find a pair of elements in our group that do not commute, this will prove that the group is non-Abelian.

Step 3: Choose Specific Matrices

Next, we select two specific matrices from \(GL(2, \mathbb{Q})\) to serve as our test case. Let matrix \(A = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\) and matrix \(B = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}\). Both matrices are invertible, hence they belong to \(GL(2, \mathbb{Q})\).

Step 4: Compute the Matrix Product AB

Calculate the product of matrices \(A\) and \(B\):\[AB = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 \cdot 1 + 1 \cdot 0 & 0 \cdot 1 + 1 \cdot 1 \ 1 \cdot 1 + 0 \cdot 0 & 1 \cdot 1 + 0 \cdot 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \ 1 & 1 \end{pmatrix}\]

Step 5: Compute the Matrix Product BA

Now calculate the product of matrices \(B\) and \(A\):\[BA = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 0 + 1 \cdot 1 & 1 \cdot 1 + 1 \cdot 0 \ 0 \cdot 0 + 1 \cdot 1 & 0 \cdot 1 + 1 \cdot 0 \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}\]

Step 6: Compare Results

Observe that\[AB = \begin{pmatrix} 0 & 1 \ 1 & 1 \end{pmatrix} \quad \text{and} \quad BA = \begin{pmatrix} 1 & 1 \ 1 & 0 \end{pmatrix}\]These two matrices are not equal, meaning \(AB eq BA\).

Step 7: Conclude Non-Abelian Nature

Since we have found a pair of matrices \(A\) and \(B\) for which \(AB eq BA\) in \(GL(2, \mathbb{Q})\), it shows that this group is non-Abelian.

Key Concepts

General Linear Group

The General Linear Group, denoted as \(GL(n, F)\), is a key concept in linear algebra, representing the set of all \(n \times n\) invertible matrices with entries from a field \(F\). Here, \(GL(2, \mathbb{Q})\) refers specifically to the group of all invertible \(2 \times 2\) matrices where the elements are rational numbers. This group is important because it encapsulates the idea of all possible transformations in a given space that can be reversed. Every matrix in this group has an inverse, which means it can be "undone."

• **Invertibility**: Every element has an inverse.
• **Closure**: The product of two invertible matrices is another invertible matrix.
• **Associativity**: Matrix multiplication is associative.
• **Identity Element**: The identity matrix serves as the neutral element.

These properties make \(GL(n, F)\) a group in algebraic terms, and in the context where \(n = 2\) and \(F = \mathbb{Q}\), it specifically examines transformations over the rational field.

Matrix Multiplication

Matrix multiplication is a fundamental operation used in many areas of mathematics and engineering. It involves taking two matrices and producing a third matrix by combining rows and columns from the two input matrices.To multiply a \(2 \times 2\) matrix \(A\) with another \(2 \times 2\) matrix \(B\), follow these steps:1. Multiply each element of the rows of \(A\) by the corresponding element of the columns of \(B\).2. Sum these products to get the elements of the resulting matrix.For example:If \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\) and \(B = \begin{pmatrix} e & f \ g & h \end{pmatrix}\), then:\[AB = \begin{pmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{pmatrix}\]This operation is not commutative, meaning in general \(AB eq BA\), which is crucial in determining the properties of matrices within the context of a group like \(GL(2, \mathbb{Q})\).

Invertible Matrices

An invertible matrix, also known as a non-singular or non-degenerate matrix, is a square matrix that possesses a unique matrix inverse. Only invertible matrices belong to the general linear group.Some key characteristics include:

• **Determinant is Non-Zero**: For a matrix to be invertible, its determinant must be non-zero.
• **Existence of an Inverse**: Given a matrix \(A\), there exists another matrix \(A^{-1}\) such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix.

The concept of invertibility is essential because it implies that a matrix transformation is reversible. This property is central to the structure of the general linear group \(GL(2, \mathbb{Q})\), where all members are invertible matrices with rational entries.

Rational Numbers

Rational numbers, represented as \(\mathbb{Q}\), are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Key characteristics:

• An example is \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q eq 0\).
• Rational numbers include integers, fractions, and finite decimals.

In the context of the general linear group \(GL(2, \mathbb{Q})\), it means that all the elements of the matrix must be rational numbers. This restriction ensures that the operations performed, such as matrix multiplication and calculations of determinants, remain within the set of rational numbers, maintaining the mathematical structure required for proofs around commutativity or non-commutativity.