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Magazine Chapter 12: Problem 5
Let \(G\) be the set of real \(2 \times 2\) matrices of the form $$ M(a)=\left(\begin{array}{ll} a & a \\ a & a \end{array}\right) $$ where \(a \neq 0\). Show that \(G\) is a group under the usual multiplication of matrices (the identity is not the usual identity matrix), and that \(G\) is isomorphic to \(\mathbb{R}^{*}\).
Short Answer
Expert verified
The set \( G \) of matrices is a group under multiplication and is isomorphic to \( \mathbb{R}^* \).
Step by step solution
01
Verify Closure Under Multiplication
The set \( G \) is closed under multiplication if the product of any two matrices from \( G \) is also a matrix in \( G \). Take two arbitrary matrices \( M(a) \) and \( M(b) \) from \( G \). Their product is:\[M(a) \cdot M(b) = \begin{pmatrix} a & a \ a & a \end{pmatrix} \begin{pmatrix} b & b \ b & b \end{pmatrix} = \begin{pmatrix} ab+ab & ab+ab \ ab+ab & ab+ab \end{pmatrix} = \begin{pmatrix} 2ab & 2ab \ 2ab & 2ab \end{pmatrix} = M(2ab).\]Since \( 2ab eq 0 \) if \( a, b eq 0 \), \( M(2ab) \) is in \( G \). Therefore, \( G \) is closed under multiplication.
02
Identify the Identity Element
The identity element is a matrix \( I \) such that for any matrix \( M(a) \) in \( G \), \( M(a) \cdot I = I \cdot M(a) = M(a) \). By observation, let \( I = M(1/2) \), which is:\[I = \begin{pmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{pmatrix}.\]Verify this by computing:\[M(a) \cdot I = \begin{pmatrix} a & a \ a & a \end{pmatrix} \begin{pmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{pmatrix} = \begin{pmatrix} a & a \ a & a \end{pmatrix} = M(a).\]So, \( I \) is indeed the identity matrix for \( G \).
03
Verify the Existence of Inverses
For each \( M(a) \) in \( G \), there should be a \( M(b) \) such that \( M(a) \cdot M(b) = I \). If we let \( M(a) = \begin{pmatrix} a & a \ a & a \end{pmatrix} \), then find \( M(b) = \begin{pmatrix} 1/(2a) & 1/(2a) \ 1/(2a) & 1/(2a) \end{pmatrix} \):\[M(a) \cdot M(b) = \begin{pmatrix} a & a \ a & a \end{pmatrix} \begin{pmatrix} 1/(2a) & 1/(2a) \ 1/(2a) & 1/(2a) \end{pmatrix} = \begin{pmatrix} 1/2 & 1/2 \ 1/2 & 1/2 \end{pmatrix} = I.\]This shows that every element has an inverse, completing the group structure.
04
Verify Associativity
Matrix multiplication is associative. For matrices \( A, B, \) and \( C \) in \( G \), \((A \cdot B) \cdot C = A \cdot (B \cdot C)\). We don't need to compute this explicitly for each pair; the associativity property of matrices guarantees it.
05
Establish Isomorphism to \( \mathbb{R}^* \)
The mapping \( \phi: G \to \mathbb{R}^* \) given by \( \phi(M(a)) = a \) is a homomorphism because for any \( M(a), M(b) \in G \):\[\phi(M(a) \cdot M(b)) = \phi(M(2ab)) = 2ab,\]and,\[\phi(M(a)) \cdot \phi(M(b)) = a \cdot b.\]Thus, \( \phi(M(a) \cdot M(b)) = 2 \cdot \phi(M(a)) \cdot \phi(M(b)). \) Furthermore, \( \phi \) is bijective, demonstrating \( G \cong \mathbb{R}^* \) up to a constant scale.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Matrix Multiplication
Matrix multiplication is a fundamental operation in linear algebra. It involves the multiplication of two matrices to produce another matrix. The process is not simply element-wise but involves a row-by-column computation. Each element in the resulting matrix is obtained by multiplying corresponding elements from the rows of the first matrix and the columns of the second matrix, then summing those products. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Also, matrix multiplication is associative, meaning that (A \cdot B) \cdot C = A \cdot (B \cdot C), where \( A, B, \) and \( C \) are matrices. This property assures us that no matter how we group the matrices during multiplication, the result will be the same. Matrix multiplication is essential in various computations, such as transformations and systems of equations, making it a critical concept to master in mathematics.
Group Theory
Group theory is an area of mathematics that studies the algebraic structures known as groups. A group is a set equipped with an operation that combines any two elements to form a third element. The operation must satisfy four key properties:
- Closure: The operation on any two elements of the group results in an element within the same group.
- Associativity: The operation is associative, meaning the grouping of operations does not affect the outcome.
- Identity Element: There exists an element in the group that, when operated with any group element, leaves that element unchanged.
- Inverse Elements: For each element in the group, there exists another element that, when operated together, results in the identity element.
Isomorphism
Isomorphism is a concept that establishes a strong structural similarity between algebraic structures, such as groups or rings. Two groups are isomorphic if there is a bijective homomorphism between them that respects the group operations. Essentially, isomorphism means that the groups have the same structure, even though their elements and operations might look different. In this exercise, we're asked to show that a group of matrices is isomorphic to \( \mathbb{R}^* \), the group of non-zero real numbers under multiplication. We demonstrate this by constructing a mapping, or homomorphism, \( \phi \), and showing it is both bijective and operation-preserving. Recognizing isomorphic structures allows mathematicians to classify and simplify problems by recognizing when two separate systems have identical behavior.
Real Matrices
Real matrices are matrices with elements that are real numbers. They are widely used in mathematics and applied fields because they represent real-world quantitative data. The set of 2x2 real matrices includes matrices like:\[M(a)=\begin{pmatrix}a & a \a & a \\end{pmatrix},\]where \(a\) is a non-zero real number. In this context, we consider the specific form of a matrix as given in the exercise. Matrices over the real numbers are crucial because they serve as representations of linear transformations in two-dimensional real space. Whether in describing geometric transformations or in solving linear systems, understanding real matrices is foundational for applications across science and engineering. Furthermore, working with real matrices provides insight into abstract mathematical concepts like vector spaces and linear mappings.
