From (3.4), show that if \(M\) is orthogonal, then det \(M=1\) or \(-1 .\) (When det \(M=1\), the transformation is called a proper rotation; when det \(M=-1\), one or all three axes have, been reflected, in addition to rotation.) Hint: Find det \(\left(M M^{T}\right) ;\) how is a determinant affected by interchanging rows and columns?

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix, such as invertibility and volume scaling. Mathematically, for a 2x2 matrix \(M = \begin{bmatrix} a & b \ \ c & d \end{bmatrix}\), its determinant is calculated as: \[ \text{det}(M) = ad - bc \] For larger matrices, the determinant is computed through recursive expansion by minors. Determinants are key in solving linear equations, analyzing matrix transformations, and understanding matrix behavior in various applications, such as physics and engineering.

A proper rotation in the context of an orthogonal matrix is a transformation that preserves both angles and orientation in a given space. When the determinant of an orthogonal matrix \(M\) is 1, it is referred to as a proper rotation. This means there is no reflection involved, and the operation purely involves rotation. Proper rotations comply fully with the preservation of geometric properties such as distances and the shape of objects. For example: \[ \text{If \ \text{det}(M) = 1,}\ \ \text{the transformation is a proper rotation.} \] Proper rotations are fundamental in 3D graphics, robot kinematics, and physical simulations to maintain the integrity of the objects being manipulated.

An identity matrix is a special type of square matrix in which all the elements on the main diagonal are 1, and all other elements are 0. It is denoted by I and serves as the multiplicative identity for matrices. For example, a 3x3 identity matrix looks like this: \[ I = \begin{bmatrix} 1 & 0 & 0 \ \ 0 & 1 & 0 \ \ 0 & 0 & 1 \end{bmatrix} \] The primary property of the identity matrix is that, when any matrix M (of compatible dimensions) is multiplied by I, the result is M itself: \[ M I = I M = M \] This matrix is crucial in simplifying various matrix operations and solving matrix equations.

The transpose of a matrix is obtained by flipping the matrix over its diagonal. This operation switches the row and column indices of the elements, transforming the \(i,j\) element of the original matrix into the \(j,i\) element of the new matrix. Mathematically, if \[ M = \begin{bmatrix} a & b \ \ c & d \end{bmatrix}, \] then the transpose \(M^T\) is \[ M^T = \begin{bmatrix} a & c \ \ b & d \end{bmatrix} \] Important properties of transposition include: \begin{itemize} \item \text{Transposing twice gives the original matrix back:} \ (M^T)^T = M \item \text{Transpose of a product:} \ (AB)^T = B^T A^T \item \text{The determinant of a matrix and its transpose are equal:} \ \text{det}(M) = \text{det}(M^T) \end{itemize} These properties are widely used in linear algebra and matrix analysis.

Matrix properties refer to the fundamental characteristics and behaviors of matrices in various operations. Some key properties include: \begin{itemize} \item \text{Commutativity:} \ Matrices generally do not commute under multiplication, meaning \(AB \e BA\). \item \text{Associativity:} \ Matrices do follow the associative property, so \(A(BC) = (AB)C\). \item \text{Distributivity:} \ Matrix multiplication is distributive over addition: \(A(B + C) = AB + AC\). \item \text{Inverse:} \ A matrix M has an inverse \(M^{-1}\) such that \(MM^{-1} = M^{-1}M = I\), if \text{det}(M) \e 0. \item \text{Trace:} \ The trace of a square matrix is the sum of its diagonal elements: \text{tr}(M) = \sum_{i} m_{ii}\text{.} \end{itemize} These properties are extensively used in solving equations, transforming coordinates, and in the broader scope of functional analysis.