/*! 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 6 If \(A=\left(\begin{array}{ll}a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 6

If \(A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)\) is skew symmetric, show that \(a=d=0\). that is the diagonal elements are zero.

Short Answer

Expert verified
For a skew-symmetric matrix, diagonal elements are zero, thus \( a = d = 0 \).

Step by step solution

01

Understand the Definition of a Skew-Symmetric Matrix

A skew-symmetric matrix is a square matrix that satisfies the condition \( A^T = -A \) where \( A^T \) is the transpose of matrix \( A \). For a \( 2 \times 2 \) matrix like \( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \), this means that when transposed and negated, the matrix remains unchanged.
02

Apply the Skew-Symmetric Property

For matrix \( A \), the transpose \( A^T = \begin{pmatrix} a & c \ b & d \end{pmatrix} \). The property \( A^T = -A \) implies \( \begin{pmatrix} a & c \ b & d \end{pmatrix} = \begin{pmatrix} -a & -b \ -c & -d \end{pmatrix} \). Equating corresponding elements gives us the equations \( a = -a \), \( b = -c \), \( c = -b \), and \( d = -d \).
03

Solve the Equations for Diagonal Elements

From the equation \( a = -a \), it follows that \( 2a = 0 \), thus \( a = 0 \). Similarly, from \( d = -d \), it follows that \( 2d = 0 \), thus \( d = 0 \). This proves that the diagonal elements of matrix \( A \) are zero.

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.

Matrix Transpose
A matrix transpose involves swapping rows for columns. For a matrix \( A \), its transpose \( A^T \) is formed by flipping it over its diagonal.
This means the first row of matrix \( A \) becomes the first column of \( A^T \), the second row becomes the second column, and so on.
If \( A \) is a \( 2 \times 2 \) matrix, like \( A=\begin{pmatrix} a & b \ c & d \end{pmatrix} \), the transpose \( A^T \) would be \( \begin{pmatrix} a & c \ b & d \end{pmatrix} \).
Transposing a matrix is a common operation in linear algebra often used to check properties such as symmetry.
  • Transposition reflects a matrix over its main diagonal.
  • It is essential in many matrix calculations and transformations.
Understanding how to transpose matrices is a fundamental step in verifying if a matrix is skew-symmetric.
Diagonal Elements
The diagonal elements of a square matrix are those elements where the row and column indices are the same.
In our example, for matrix \( A=\begin{pmatrix} a & b \ c & d \end{pmatrix} \), the diagonal elements are \( a \) and \( d \).

For skew-symmetric matrices, these diagonal elements have a unique property: they are always zero.
This is because, by definition, a skew-symmetric matrix \( A \) satisfies \( A^T = -A \).
Applying this to the diagonal elements leads to equations like \( a = -a \), which simplifies to \( 2a = 0 \), thus \( a = 0 \).

  • Diagonal elements are critical in determining a matrix's properties.
  • The zero-diagonal property helps simplify many computations and is a telltale marker of skew-symmetry.
Knowing these characteristics aids in quickly identifying when a matrix is skew-symmetric.
Equations in Matrices
Equations in matrices often emerge when identifying or proving specific properties, such as skew-symmetry.
In our exercise, we use matrix equations to equate the original matrix to its negative transpose.
For the skew-symmetric matrix \( A=\begin{pmatrix} a & b \ c & d \end{pmatrix} \), the equation \( A^T = -A \) results in several smaller equations:
  • \( a = -a \), which implies \( 2a = 0 \) thus making \( a = 0 \)
  • \( b = -c \)
  • \( c = -b \)
  • \( d = -d \), which implies \( 2d = 0 \) thus making \( d = 0 \)
These equations show how the diagonal elements become zero and how the off-diagonal entries relate to each other with a sign change.

Utilizing such equations allows us to solve and explore matrices and provides insights into their structure and characteristics.

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

Determine the type of matrix obtained when two liagonal matrices are multiplied together.

If \(\mathbf{a}=7 \mathbf{i}+11 \mathbf{j}-2 \mathbf{k}\) and \(\mathbf{b}=6 \mathbf{i}-3 \mathbf{j}+\underline{\mathbf{k} \text { find } \mathbf{a} \times \mathbf{b} \text {. }}\)

By expressing the following equations in matrix form and finding an inverse matrix, solve. (a) \(4 x-2 y=14\) \(2 x+y=5\) (b) \(\begin{aligned} 2 x-2 y &=0 \\ x+3 y &=-8 \end{aligned}\) (c) \(8 x+3 y=59\) \(-2 x+y=-13\)

Find \(\mid \begin{array}{cc}\cos \omega t & \sin \text { ott } \\ -\sin \omega t & \cos \omega t\end{array}\).

If \(A=\left(\begin{array}{rr}3 & 0 \\ -1 & 4\end{array}\right)\) and \(B=\left(\begin{array}{ll}7 & 8 \\ 4 & 3\end{array}\right)\) find \(|A B|,|B A| .\)
See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Nuclear Physics

Read Explanation

Oscillations

Read Explanation

Classical Mechanics

Read Explanation

Quantum Physics

Read Explanation

Translational Dynamics

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.