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Chapter 2: Problem 64

Prove that if \(A\) is an \(n \times n\) matrix, then \(A-A^{T}\) is skewsymmetric.

Short Answer

Expert verified
The matrix \(A - A^{T}\) is always skewsymmetric for any square matrix \(A\).

Step by step solution

01

Calculate \(A - A^{T}\)

The result is a new matrix where for each element, we subtract the corresponding element in \(A^{T}\) from the same positioned element in \(A\). Let's call the resultant matrix \(B\). So, \(B_{ij}=A_{ij}-A_{ji}\), where \(B_{ij}\) is the element in the \(i\)-th row and \(j\)-th column of \(B\).
02

Transpose \(B\)

The transpose of \(B\), \(B^{T}\) is obtained by interchanging the row and column indices of each element. That is, if \(B_{ij}\) is an element of \(B\), then its corresponding element in \(B^{T}\) is \(B_{ji}\). Thus, \(B_{ji}=A_{ji}-A_{ij}\) for \(B^{T}\).
03

Compare \(B\) and \(B^{T}\)

From the definitions of \(B\) and \(B^{T}\), it can be seen that for any element \(B_{ij}\), its corresponding element in \(B^{T}\) is the negative of it. So, \(B_{ij}=-B_{ji}\), which is the property of a skewsymmetric matrix.

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Key Concepts

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

Understanding Linear Algebra
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear mappings, and systems of linear equations. It's the foundation for many areas of mathematics and applications in physics, engineering, computer science, and even social sciences.

When studying linear algebra, you'll encounter matrices, which are arrays of numbers that can represent data, transformations, and more. Matrices come in various types, with properties that enable us to solve complex problems easily. For instance, understanding the nature of a matrix, such as whether it鈥檚 skewsymmetric, can help us predict its behavior in equations and transformations.

A skewsymmetric matrix, also sometimes called an antisymmetric matrix, is a square matrix that's equal to the negation of its transpose, which leads us to our next key concept.
Matrix Transpose in a Nutshell
The transpose of a matrix is essentially a new matrix obtained by flipping it over its diagonal. Formally, if you have a matrix 'A' with elements denoted as \(A_{ij}\), where \(i\) is the row index and \(j\) is the column index, the transpose of 'A', denoted as \(A^T\), will have its elements flipped such that \(A_{ij}\) in 'A' becomes \(A_{ji}\) in \(A^T\).

Transposing is a crucial operation in linear algebra, as it can change the properties of a matrix and affect the outcome of matrix multiplications and other operations. Understanding how to work with transposes is fundamental for anyone delving into the depths of linear algebra.
The Role of Proofs in Linear Algebra
Linear algebra is rife with structures that often require rigorous proof to understand and justify their properties and relationships. Proofs in linear algebra confirm the truths about vectors, spaces, and matrix characteristics, making these truths universally accepted and applicable.

For example, in the exercise provided, we prove a matrix 'B', derived from subtracting the transpose of a matrix 'A' from 'A' itself, is skewsymmetric. This proof methodically tests the definition of a skewsymmetric matrix against the elements of matrix 'B'. Through this kind of proof, we demonstrate the fundamental nature of 'B', confirming the property \(B_{ij} = -B_{ji}\) for all \(i, j\), which is essential knowledge for anyone studying the subject.

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Most popular questions from this chapter

let \(A, B,\) and \(C\) be $$A=\left[\begin{array}{rrr} 1 & 2 & -3 \\ 0 & 1 & 2 \\ -1 & 2 & 0 \end{array}\right]$$ $$B=\left[\begin{array}{rrr} -1 & 2 & 0 \\ 0 & 1 & 2 \\ 1 & 2 & -3 \end{array}\right]$$ $$C=\left[\begin{array}{rrr} 0 & 4 & -3 \\ 0 & 1 & 2 \\ -1 & 2 & 0 \end{array}\right]$$ Find an elementary matrix \(E\) such that \(E C=A\)

Prove that if \(A, B,\) and \(C\) are square matrices and \(A B C=I\), then \(B\) is invertible and \(B^{-1}=C A\)

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & -1 & 3 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & -1 \end{array}\right]$$

Under what conditions will the diagonal matrix $$A=\left[\begin{array}{ccccc} a_{11} & 0 & 0 & \cdots & 0 \\ 0 & a_{22} & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 & 0 & \cdots & a_{n n} \end{array}\right]$$ be invertible? If \(A\) is invertible, find its inverse.

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 1 & 3 & 4 \end{array}\right]$$
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