Question

Given that matrix \(M\) is a member of the multiplicative group \(G L(3, \mathbf{R})\), determine, for each of the following additional constraints on \(M\) (applied separately), whether the subset satisfying the constraint is a subgroup of \(G L(3, \mathbf{R})\) : (a) \(\mathrm{M}^{T}=\mathrm{M}\) (b) \(\mathrm{M}^{\bar{T}} \mathrm{M}=\mathrm{I}\); (c) \(|\mathrm{M}|=1\);

Short answer

(a) No, (b) Yes, (c) Yes

Step-by-step solution

Definition of Subgroup

To determine if a subset is a subgroup, it must satisfy the following criteria: closure under multiplication, existence of the identity element, and existence of inverses within the subset.

Analyze Condition (a) - Symmetric Matrix

For a matrix M such that \(\text{M}^T = \text{M}\), if \(\text{M}_1, \text{M}_2\) are symmetric, then \((\text{M}_1 \text{M}_2})^T = \text{M}_2^T \text{M}_1^T = \text{M}_2 \text{M}_1\). Symmetric matrices are not closed under multiplication. Hence, the subset is not a subgroup.

Analyze Condition (b) - Orthogonal Matrix

For a matrix \(\text{M}\) that satisfies \(\text{M}^{\bar{T}} \text{M} = \text{I}\), orthogonal matrices are closed under multiplication, contain the identity matrix and inverses of orthogonal matrices are also orthogonal. Hence, the subset is a subgroup.

Analyze Condition (c) - Determinant Equals 1

For matrices \(\text{M}\) such that \(|\text{M}| = 1\), matrices with a determinant of 1 are closed under multiplication, contain the identity matrix (which has a determinant of 1), and inverses of matrices with a determinant of 1 also have a determinant of 1. Hence, the subset is a subgroup.

Key concepts

Multiplicative Group

A multiplicative group is a set equipped with a multiplication operation that satisfies certain properties. In this context, we are dealing with the general linear group, denoted as \(GL(3, \mathbb{R})\), which consists of all 3x3 invertible matrices with real entries. This group has three important properties:

- **Closure under multiplication**: The product of any two invertible matrices is also an invertible matrix.

- **Identity element**: There is a special matrix, the identity matrix \(I\), which when multiplied with any matrix \(M\), leaves \(M\) unchanged: \(I \cdot M = M \cdot I = M\).

- **Existence of inverses**: For every matrix in the group, there is another matrix that, when multiplied together, gives the identity matrix: \(M \cdot M^{-1} = M^{-1} \cdot M = I\).

Understanding these properties helps us analyze subgroups, which must also satisfy them. Now, let's look at specific kinds of matrices to see if they form subgroups within \(GL(3, \mathbb{R})\).

Symmetric Matrix

A symmetric matrix is a matrix that is equal to its transpose: \(M^T = M\). Transposing a matrix means swapping its rows and columns. For example, if matrix \(M\) has elements \(a_{ij}\), then \(M^T\) will have elements \(a_{ji}\).

To determine if symmetric matrices form a subgroup of \(GL(3, \mathbb{R})\), we need to check the subgroup criteria. Unfortunately, symmetric matrices are **not closed under multiplication**. That means if you multiply two symmetric matrices, the result may not be symmetric.

Given this, we cannot form a subgroup from symmetric matrices within \(GL(3, \mathbb{R})\). Even though symmetric matrices are an interesting class of matrices, the failure of closure under multiplication disqualifies them from being a subgroup.

Orthogonal Matrix

An orthogonal matrix \(M\) satisfies the condition \(M^T M = I\), where \(I\) is the identity matrix. This means its transpose is also its inverse. Orthogonal matrices have some special properties:

- **Preservation of length**: They preserve the lengths of vectors, making them very useful in applications involving rotations and reflections.

- **Orthogonality of columns**: The columns of an orthogonal matrix are orthogonal unit vectors.

Orthogonal matrices **do form a subgroup** within \(GL(3, \mathbb{R})\) because they satisfy closure under multiplication, contain the identity element, and contain inverses that are also within the group. This makes them an important and useful class of matrices in various mathematical and applied contexts.

Determinant

The determinant of a matrix is a special number that provides important information about the matrix, such as its invertibility and volume scaling factor in transformations. The determinant is denoted as \(|M|\).

For our context, we focus on matrices with a determinant of 1, often called **special linear matrices** and denoted as \(SL(3, \mathbb{R})\). These matrices form a subgroup within \(GL(3, \mathbb{R})\), as they satisfy:

- **Closure under multiplication**: If \(M_1\) and \(M_2\) both have a determinant of 1, then \(M_1M_2\) also has a determinant of 1.

- **Identity element**: The identity matrix has a determinant of 1.

- **Existence of inverses**: If \(M\) has a determinant of 1, its inverse \(M^{-1}\) also has a determinant of 1.

This subset is a subgroup and is widely used in linear algebra and its applications due to its interesting and useful properties.