Question

Let \(P\) be an orthogonal matrix. a. Show that \(\operatorname{det} P=1\) or \(\operatorname{det} P=-1\). b. Give \(2 \times 2\) examples of \(P\) such that \(\operatorname{det} P=1\) and \(\operatorname{det} P=-1\) c. If \(\operatorname{det} P=-1\), show that \(I+P\) has no inverse. \(\left[\right.\) Hint \(\left.: P^{T}(I+P)=(I+P)^{T} .\right]\) d. If \(P\) is \(n \times n\) and det \(P \neq(-1)^{n}\), show that \(I-P\) has no inverse. \(\left[\right.\) Hint: \(\left.P^{T}(I-P)=-(I-P)^{T} .\right]\)

Short answer

Orthogonal matrices have \(\operatorname{det} P = \pm 1\). Examples: \(\operatorname{det} P = 1\) for identity, \(-1\) for permutation matrix. \(I+P\) is non-invertible for \(\operatorname{det} P = -1\), \(I-P\) is non-invertible if \(\operatorname{det} P \neq (-1)^n\).

Step-by-step solution

Definition of an Orthogonal Matrix

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors. This means that for a matrix \( P \), it satisfies \( P^T P = I \) where \( P^T \) is the transpose of \( P \) and \( I \) is the identity matrix.

Determinant Property of Orthogonal Matrices

For an orthogonal matrix \( P \), the determinant has the property that \( \det(P^T) = \det(P) \) and \( \det(P^T P) = \det(I) = 1 \). This implies \( \det(P)^2 = 1 \), leading to \( \det(P) = \pm 1 \).

Examples of 2x2 Orthogonal Matrices

Consider the matrix \( P = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). It is orthogonal and \( \det(P) = 1 \). Another example, \( P = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \), is also orthogonal and \( \det(P) = -1 \).

Showing \( I+P \) is Non-Invertible When \( \det(P) = -1 \)

If \( \det(P) = -1 \), consider \( P^T (I+P) = (I+P)^T \). Since \( P^T = P^{-1} \) for orthogonal matrices, this implies \( P^{-1}(I+P) = (I+P)^T \). Taking the determinant, \( \det(P^{-1})\det(I+P) = \det((I+P)^T) \). But if \( \det(I+P) eq 0 \), it should lead to an inconsistency with \( \det(P) = -1 \), thus \( I+P \) is singular.

Condition for \( I-P \) Non-Invertibility

Given \( P^T (I-P) = -(I-P)^T \), and det \( P eq (-1)^n \), consider \( (I-P)^T = -(I-P) \). If \( I-P \) were invertible, we can multiply both sides by this inverse leading to a contradiction unless \( det(I-P) = 0 \), which signifies that \( I-P \) is non-invertible.

Key concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It helps us understand many properties of the matrix, such as whether it is invertible. For an orthogonal matrix \( P \), an interesting property holds: the determinant can only be \( 1 \) or \( -1 \).

Why is this the case? Since \( P \) is orthogonal, it satisfies \( P^T P = I \), where \( P^T \) is the transpose of \( P \) and \( I \) is the identity matrix. When we take the determinant of both sides, we get \( \det(P^T P) = \det(I) = 1 \). This results in \( \det(P)^2 = 1 \), leading to \( \det(P) = \pm 1 \).

Determinants are crucial for understanding matrices because they provide insight into the matrix's characteristics, such as rotation and reflection in geometry.

Matrix Invertibility

Matrix invertibility refers to the ability to find a matrix's inverse. A matrix is invertible if there is another matrix that, when multiplied with it, yields the identity matrix. However, not all matrices have an inverse.

For example, if an orthogonal matrix \( P \) has \( \det(P) = 1 \), it is usually invertible. However, if \( \det(P) = -1 \), things change. In the case of \( I+P \), where \( \det(P) = -1 \), the matrix is non-invertible because the resulting determinant calculation leads to an inconsistency, showing \( \det(I+P) \) equals zero. Similarly, if \( \det(P) eq (-1)^n \) for an \( n \times n \) matrix, \( I-P \) also becomes non-invertible.

This concept helps in determining when a matrix can perform operations that rely on its inverse.

2x2 Matrices

2x2 matrices are a simple yet powerful form of matrices that often serve as introductory examples for learning matrix concepts. A 2x2 matrix consists of two rows and two columns, allowing various operations and applications.

For orthogonal 2x2 matrices, we can have:\[ P = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
This matrix has a determinant of \( 1 \) and represents identity or no change.

Another example is:\[ P = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \]
This matrix swaps the values and has a determinant of \( -1 \).

These examples show how 2x2 matrices can represent basic geometric transformations like identity and reflections.