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Chapter 1: Problem 17

Decide whether the given matrix is symmetric. $$\left[\begin{array}{rrr}0 & 1 & 2 \\ 1 & 5 & -6 \\ 2 & 6 & 6\end{array}\right]$$

Short Answer

Expert verified
The matrix is not symmetric.

Step by step solution

01

Understand Symmetric Matrix Definition

A symmetric matrix is a square matrix that is equal to its transpose. In simpler terms, a matrix \( A \) is symmetric if \( A = A^T \). This means that the elements of the matrix satisfy the condition that \( a_{ij} = a_{ji} \) for all indices \( i \) and \( j \).
02

Compare Elements Across Main Diagonal

Examine the elements across the main diagonal of the given matrix:\[\begin{bmatrix}0 & 1 & 2 \1 & 5 & -6 \2 & 6 & 6\end{bmatrix}\]We need to check if \( a_{ij} = a_{ji} \) for all elements.
03

Check \( a_{12} \) and \( a_{21} \)

\( a_{12} = 1 \) and \( a_{21} = 1 \). These elements are equal, satisfying the symmetry condition for these positions.
04

Check \( a_{13} \) and \( a_{31} \)

\( a_{13} = 2 \) and \( a_{31} = 2 \). These elements are also equal, satisfying the symmetry condition for these positions.
05

Check \( a_{23} \) and \( a_{32} \)

\( a_{23} = -6 \) and \( a_{32} = 6 \). These elements are not equal, showing that the matrix is not symmetric.

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Key Concepts

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

Symmetric Matrix
A symmetric matrix is a fascinating and comprehensible concept in linear algebra. At its core, a symmetric matrix is a square matrix that mirrors itself along its main diagonal.
This means each element on one side of the diagonal is equal to the corresponding element on the other side.
For a matrix to be considered symmetric, it must satisfy the criteria:
  • It should be a square matrix. This means the number of rows and columns must be the same.
  • The elements must satisfy the condition: \( a_{ij} = a_{ji} \).
To decide if a matrix is symmetric, check each pair of corresponding elements across the diagonal. If all of them are equal, the matrix is symmetric.This property has important implications in various fields, such as physics and engineering, because of its simplicity and elegance.
Transpose of a Matrix
The transpose of a matrix is a fundamental operation in matrix algebra.
It involves flipping the matrix over its main diagonal, swapping the row and column indices of each element.
If you have a matrix \( A \), its transpose, denoted as \( A^T \), would have the elements
  • from the first row become the first column
  • from the second row become the second column, and so on
For a given matrix \( A = \begin{bmatrix} 0 & 1 & 2 \ 1 & 5 & -6 \ 2 & 6 & 6 \end{bmatrix} \), its transpose, \( A^T \), would be:\[A^T = \begin{bmatrix} 0 & 1 & 2 \ 1 & 5 & 6 \ 2 & -6 & 6 \end{bmatrix} \]This operation is used in various applications, like solving linear equations, and is crucial for defining symmetric matrices.
Matrix Elements Equality
Matrix elements equality is a key concept for determining matrix symmetry.
It focuses on checking whether specific elements in the matrix are equal to their transposed counterparts.
The equality condition \( a_{ij} = a_{ji} \) must hold through all elements for a matrix to be symmetric.When examining the given matrix:\[\begin{bmatrix} 0 & 1 & 2 \ 1 & 5 & -6 \ 2 & 6 & 6 \end{bmatrix}\]We compare pairs like:
  • \(a_{12} = 1\) with \(a_{21} = 1\)
  • \(a_{13} = 2\) with \(a_{31} = 2\)
  • \(a_{23} = -6\) with \(a_{32} = 6\)
Here, we notice that \(a_{23}\) and \(a_{32}\) are not equal. This inequality confirms that the matrix is not symmetric. Understanding these element pairs helps quickly identify asymmetries in matrices for applications where symmetry is desired.

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Most popular questions from this chapter

Determine the values of \(a\) for which the system has no solutions, exactly one solution, or infinitely many solutions. $$\begin{aligned}x+2 y-& \quad 3 z=\quad4 \\\3 x-y+\quad& 5 z=\quad 2\\\4 x+y+\left(a^{2}-14\right) z &=a+2 \end{aligned}$$

Let \(A\) be any \(m \times n\) matrix and let \(\theta\) be the \(m \times n\) matrix each of whose entries is zero. Show that if \(k A=0,\) then \(k=0\) or \(A=0.\)

Find all values of the unknown constant(s) in order for \(A\) to be symmetric. $$A=\left[\begin{array}{cc} 4 & -3 \\ a+5 & -1 \end{array}\right]$$

Use the inversion algorithm to find the inverse of the given matrix, if the inverse exists. $$\left[\begin{array}{rr} -1 & 3 \\ 3 & -2 \end{array}\right]$$

(a) Show that if \(A\) has a row of zeros and \(B\) is any matrix for which \(A B\) is defined, then \(A B\) also has a row of zeros. (b) Find a similar result involving a column of zeros.
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