/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find a \(2 \times 2\) matrix wit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 23

Find a \(2 \times 2\) matrix with determinant I that is not an orthogonal matrix.

Short Answer

Expert verified
A 2x2 matrix with determinant 1 that is not orthogonal is \(\begin{pmatrix} 2 & 1 \\ 0 & \frac{1}{2} \end{pmatrix}\).

Step by step solution

01

Understand Matrix Types

We need to find a matrix with specific properties 1. **Determinant is I**: This means the determinant should be 1 (as "I" usually represents 1 in this context). 2. **Not Orthogonal**: This means the matrix does not satisfy that its transpose equals its inverse, i.e., \(A^T A eq I\), where \(I\) is the identity matrix.
02

Set Up the General Form of a 2x2 Matrix

A typical \(2 \times 2\) matrix \(A\) is represented as:\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]The determinant is calculated as \(ad - bc\). To have \(\det(A) = 1\), we need:\[ad - bc = 1\].
03

Choose Elements of the Matrix

Select values for \(a, b, c,\) and \(d\) such that \(ad - bc = 1\) and not all rows and columns are orthogonal. Let's choose: \[a = 2, \quad b = 1, \quad c = 0, \quad d = \frac{1}{2}\]
04

Verify the Determinant

Calculate the determinant of the chosen matrix:\[\det(A) = (2)(\frac{1}{2}) - (0)(1) = 1 - 0 = 1\]This satisfies the condition \(\det(A) = 1\).
05

Check Orthogonality

To verify that the matrix is not orthogonal, the product \(A^T A\) should not equal the identity. For \(A = \begin{pmatrix} 2 & 1 \ 0 & \frac{1}{2} \end{pmatrix}\),first find \(A^T = \begin{pmatrix} 2 & 0 \ 1 & \frac{1}{2} \end{pmatrix}\).Then calculate \(A^T A:\)\[A^T A = \begin{pmatrix} 2 & 0 \ 1 & \frac{1}{2} \end{pmatrix} \begin{pmatrix} 2 & 1 \ 0 & \frac{1}{2} \end{pmatrix} = \begin{pmatrix} 4 & 2 \ 2 & \frac{5}{4} \end{pmatrix} eq \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\]Thus, \(A\) is not orthogonal.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Determinant
The determinant of a matrix is a special number that can be calculated from its elements. For a simple \(2 \times 2\) matrix \(A\) represented as:
  • \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \)
The determinant \( \det(A) \) is given by the formula:
  • \( \det(A) = ad - bc \)
The value of the determinant provides insights into several properties of the matrix:
  • If the determinant is zero, the matrix is singular, meaning it does not have an inverse.
  • If the determinant is non-zero, then the matrix has an inverse, and its rank is full (equal to the number of rows or columns).
  • The determinant also indicates the scaling factor of the linear transformation described by the matrix.
In this exercise, we are looking for a \(2 \times 2\) matrix with a determinant of 1. This means it performs a transformation that preserves volume (it neither shrinks nor expands space).
Orthogonal Matrix
An orthogonal matrix is a special kind of square matrix that has some very interesting properties:
  • An orthogonal matrix \(A\) satisfies \(A^T A = I\), where \(A^T\) is the transpose of \(A\), and \(I\) is the identity matrix.
  • Multiplying by an orthogonal matrix represents a rotation or reflection in space without changing the length of vectors (i.e., it preserves angles and distances).
  • The inverse of an orthogonal matrix is its transpose, making calculations straightforward.
Orthogonal matrices have a determinant of either 1 or -1. In this exercise, we are tasked with finding a \(2 \times 2\) matrix with a determinant of 1 that is not orthogonal. This means the product \(A^T A\) must not be the identity matrix.
Transpose
The transpose of a matrix is obtained by flipping it over its diagonal. For a matrix \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the transpose, denoted \(A^T\), becomes:
  • \( A^T = \begin{pmatrix} a & c \ b & d \end{pmatrix} \)
Here's why the transpose is important:
  • It is used in checking the orthogonality of a matrix. If the product of a matrix and its transpose equals the identity matrix, it is orthogonal.
  • It plays a crucial role in various mathematical applications, including solving systems of equations and statistical analyses.
  • Transpose helps to express and solve problems where changing the configuration or perspective of a dataset is needed.
Understanding transposes helps in verifying or disproving properties like orthogonality in matrices, as seen in this exercise where we confirmed that the matrix is not orthogonal by comparing \(A^T A\) with \(I\).

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find an orthonormal basis for \(\mathrm{sp}\left(1, e^{x}\right)\) for \(0 \leq x \leq 1\) if the inner product is defined by \(\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x\).

Determine whether the given linear transformation is orthogonal. \(T: \mathrm{R}^{3} \rightarrow \mathrm{R}^{3}\) defined by \(T([x, y, z])=[x, y, 0]\)

Find the projection matrix for the given subspace, and find the projection of the indicated vector on the subspace. \([1,2,1,3]\) on sp( \([i, 2,1,1],[-1,1,0,-1])\) in \(\mathbf{R}^{4}\)

Find the least-squares fit to the given data by a linear function \(f(x)=r_{0}+r_{1} x\). Graph the linear function and the data points. $$ (0,0),(1,1),(2,3),(3,8) $$

Find an orthonormal basis for the plane \(2 x+3 y+z=0\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.