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Chapter 2: Problem 5

a. Is \(I\) an elementary matrix? Explain. b. Is 0 an elementary matrix? Explain.

Short Answer

Expert verified
a. Yes, \(I\) is an elementary matrix. b. No, 0 is not an elementary matrix.

Step by step solution

01

Define an Elementary Matrix

An elementary matrix is obtained by performing a single elementary row operation on an identity matrix. There are three types of elementary matrices: 1) swapping two rows, 2) multiplying a row by a non-zero scalar, and 3) adding a multiple of one row to another row.
02

Analyze Identity Matrix (I)

The identity matrix, denoted as \(I\), is a square matrix with ones on the diagonal and zeros elsewhere. It represents the matrix equivalent of the number 1 and does not change a vector when it is multiplied. Since an identity matrix can be seen as an identity matrix without modification, it is an example of an elementary matrix.
03

Analyze Zero Matrix (0)

A zero matrix is a matrix consisting exclusively of zeros. It does not result from any elementary row operation applied to an identity matrix. Instead, any operation applied to an identity matrix that results in a zero matrix would either require multiplying all rows by zero or make a series of invalid row operations. Thus, a zero matrix is not an elementary matrix.

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Key Concepts

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

Identity Matrix
An identity matrix is like the magical chameleon of matrices. It looks plain, but in matrix operations, it's incredibly special. An identity matrix, often denoted as \(I\), is a square matrix with all the diagonal elements equal to one, and every other element is zero. This structure creates a powerful effect in multiplication because when you multiply any matrix by an identity matrix, the original matrix remains unchanged. It's like multiplying a number by one; the value stays the same.

  • Structure: All ones on the diagonal, zeros everywhere else.
  • Behavior: It acts like the number one in matrix multiplication.
In the context of elementary matrices, an identity matrix is the stepping-off point. Since it doesn't require any modification to become an elementary matrix, it can naturally be considered an example of one. The identity matrix is crucial for solving equations and understanding transformations in linear algebra.
Zero Matrix
The zero matrix is the ultimate minimalist in the world of matrices. As its name implies, a zero matrix is filled entirely with zeros, no matter its size or shape. Symbolized often by \(0\), it behaves uniquely in arithmetic operations.

  • Structure: Every element is zero.
  • Behavior: Any matrix multiplied by a zero matrix results in a zero matrix.
Unlike the identity matrix, the zero matrix does not fit into the category of elementary matrices because you cannot create it through a single elementary row operation on an identity matrix. Any attempt to turn an identity matrix into a zero matrix would involve multiplying rows by zero, which isn’t a valid single operation in the process of forming elementary matrices. Therefore, while the zero matrix is fundamental in different aspects of linear algebra, it isn't an elementary matrix.
Elementary Row Operations
Elementary row operations are like the toolset for reshaping matrices. These operations transform matrices in straightforward ways that help solve equations or simplify complex matrix structures. There are three main types of elementary row operations:

  • Swapping two rows.
  • Multiplying a row by a non-zero scalar.
  • Adding a multiple of one row to another row.
The intriguing part about these operations is their ability to maintain the essence of a matrix while transforming its appearance. These operations are what generate elementary matrices when applied to an identity matrix. Each type of operation should be practiced until it feels second nature, as they are foundational for working with larger systems in linear algebra. Understanding these operations will also enhance your comprehension of concepts like matrix inverses and determinants.

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Most popular questions from this chapter

Let \(A\) and \(B\) denote \(n \times n\) invertible matrices. a. Show that \(A^{-1}+B^{-1}=A^{-1}(A+B) B^{-1}\). b. If \(A+B\) is also invertible, show that \(A^{-1}+B^{-1}\) is invertible and find a formula for \(\left(A^{-1}+B^{-1}\right)^{-1}\).

In each case assume that \(A\) is a square matrix that satisfies the given condition. Show that \(A\) is invertible and find a formula for \(A^{-1}\) in terms of \(A\). a. \(A^{3}-3 A+2 I=0\). b. \(A^{4}+2 A^{3}-A-4 I=0\).

Let \(Q_{0}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be reflection in the \(x\) axis, let \(Q_{1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be reflection in the line \(y=x,\) let \(Q_{-1}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be reflection in the line \(y=-x,\) and let \(R_{\frac{\pi}{2}}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\) be counterclockwise rotation through \(\frac{\pi}{2}\) a. Show that \(Q_{1} \circ R_{\frac{\pi}{2}}=Q_{0}\). b. Show that \(Q_{1} \circ Q_{0}=R_{\frac{\pi}{2}}\). c. Show that \(R_{\frac{\pi}{2}} \circ Q_{0}=Q_{1}\). d. Show that \(Q_{0} \circ R_{\frac{\pi}{2}}=Q_{-1}\).

Let \(S: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) and \(T: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) be linear transformations with matrices \(A\) and \(B\) respectively. a. Show that \(B^{2}=B\) if and only if \(T^{2}=T\) (where \(T^{2}\) means \(T \circ T\) ). b. Show that \(B^{2}=I\) if and only if \(T^{2}=1_{\mathbb{R}^{n}}\) c. Show that \(A B=B A\) if and only if \(S \circ T=T \circ S\).

In each case find an invertible matrix \(U\) such that \(U A=B\), and express \(U\) as a product of elementary matrices. a. \(A=\left[\begin{array}{rrr}2 & 1 & 3 \\ -1 & 1 & 2\end{array}\right], B=\left[\begin{array}{rrr}1 & -1 & -2 \\ 3 & 0 & 1\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}2 & -1 & 0 \\ 1 & 1 & 1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & 1 \\ 2 & -1 & 0\end{array}\right]\)
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