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Chapter 1: Problem 38

Prove that, if \(A\) is a square matrix, then the matrix \(A+A^{T}\) is symmetric.

Short Answer

Expert verified
The matrix \(A + A^{T}\) is symmetric because \((A + A^{T})^{T} = A + A^{T}\).

Step by step solution

01

Understand Matrix Transposition

Matrix transposition involves converting a matrix such that its rows become columns. If we have a matrix \(A\), the transpose of \(A\) is represented as \(A^{T}\). For a symmetric matrix, the property \(A = A^{T}\) holds true.
02

Define the Expression to Prove

We need to show that the matrix \(A + A^{T}\) is symmetric. This requires proving that \((A + A^{T})^{T} = A + A^{T}\).
03

Apply the Transpose Property to the Expression

Use the property of transpose on the sum of two matrices: \((B + C)^{T} = B^{T} + C^{T}\). Apply this property to \((A + A^{T})^{T}\), which gives us \(A^{T} + (A^{T})^{T}\).
04

Simplify the Expression Using Properties of Transpose

Recognize that \((A^{T})^{T} = A\). Therefore, \(A^{T} + (A^{T})^{T}\) simplifies to \(A^{T} + A\).
05

Compare the Simplified Expression to the Original

Observe that \(A^{T} + A\) is the same as \(A + A^{T}\) since addition is commutative. This leads to \((A + A^{T})^{T} = A + A^{T}\), satisfying the condition for symmetry.

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Key Concepts

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

Matrix Transposition
Matrix transposition is a fundamental operation in linear algebra. It involves swapping the rows and columns of a matrix. For a matrix denoted as \( A \), its transpose is represented by \( A^{T} \).
  • Each element \( a_{ij} \) in the original matrix \( A \) transforms into \( a_{ji} \) in \( A^{T} \).
  • If \( A \) is of order \( m \times n \), then \( A^{T} \) is of order \( n \times m \).
  • Transposing a matrix twice returns the original matrix, i.e., \((A^{T})^{T} = A\).
Matrix transposition is useful in various applications such as solving linear equations and transformations. It's essential to understand this concept, as it plays a big role in proving properties related to symmetric matrices.
Symmetric Matrix
A symmetric matrix is a special kind of square matrix. It is equal to its own transpose, meaning for a matrix \( A \): \( A = A^{T} \).
  • To be symmetric, a matrix must be square, i.e., have the same number of rows and columns.
  • In practical terms, the element at the \( i \)-th row and \( j \)-th column must be equal to the element at the \( j \)-th row and \( i \)-th column.
  • For example, the matrix \(\begin{bmatrix} 1 & 2 \ 2 & 1 \end{bmatrix}\) is symmetric, as swapping rows and columns does not change it.
Symmetric matrices appear frequently in engineering and physics, particularly in problems involving quadratic forms and eigenvalues.
Transpose Property
The transpose property is a useful rule when working with matrices. It states how operations with matrices behave under transposition. The property \((B+C)^{T} = B^{T} + C^{T}\) is particularly important.
  • It tells us that the transpose of the sum of two matrices is the same as the sum of their transposes.
  • This property can simplify calculations and is critical in proofs involving matrix symmetry.
  • Another important transpose property is that \((BC)^{T} = C^{T}B^{T}\) for any matrices \( B \) and \( C \).
The transpose properties help make sense of complex matrix operations and support diverse mathematical proofs.
Square Matrix
A square matrix is a matrix with the same number of rows and columns. It's denoted as an \( n \times n \) matrix.
  • Square matrices are central in linear algebra, as many important concepts, like determinants and eigenvalues, apply to them.
  • Properties such as diagonalizability and being able to define an inverse are unique to square matrices.
  • Square matrices can be symmetric, skew-symmetric, diagonal, or identity matrices, depending on their specific properties.
Understanding square matrices is essential, as they form the basis of many mathematical and real-world applications, influencing how systems of linear equations are solved.

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Most popular questions from this chapter

In Exercises 30-37, describe all possible values for the unknowns \(x_{i}\) so that the matrix equation is valid. \(2\left[\begin{array}{ll}x_{1} & x_{2}\end{array}\right]-\left[\begin{array}{ll}4 & 7\end{array}\right]=\left[\begin{array}{ll}-2 & 11\end{array}\right]\)

In Exercises 1-i6, let $$ A=\left[\begin{array}{rrr} -2 & 1 & 3 \\ 4 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & 1 & -2 \\ 5 & -1 & 3 \end{array}\right], \quad C=\left[\begin{array}{rr} 2 & -1 \\ 0 & 6 \\ -3 & 2 \end{array}\right], \quad \text { and } \quad D=\left[\begin{array}{rr} -4 & 2 \\ 3 & 5 \\ -1 & -3 \end{array}\right] \text {. } $$ Compute the indicated quantity, if it is defined. $$ C-D $$

In Exercises 1-6, reduce the matrix to (a) row-echelon form, and (b) reduced row-echelor: form. Answers to \((a)\) are not unique, so your answer may differ from the one at the back of the text. \(\left[\begin{array}{rrrr}0 & 2 & -1 & 3 \\ -1 & 1 & 2 & 0 \\ 1 & 1 & -3 & 3 \\\ 1 & 5 & 5 & 9\end{array}\right]\)

In Exercises 7-12, describe all solutions of a linear system whose corresponding augmented matrix can be row-reduced to the given matrix. If requested, also give the indicated particular solution, if it exists. \(\left[\begin{array}{rrr|r}1 & -1 & 2 & 3 \\ 0 & 1 & 4 & 2\end{array}\right]\), solution with \(x_{3}=2\)

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. The value of \(y\) such that \([-3, y, 10]\) is perpendicular to \(\mathrm{u}\)
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