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Chapter 3: Problem 5

Prove that \(E\) is an elementary matrix if and only if \(E^{t}\) is.

Short Answer

Expert verified
We prove that \(E\) is an elementary matrix if and only if \(E^t\) is by analyzing the behavior of the transpose with respect to the three types of elementary row operations: 1. Swapping two rows in \(E\) corresponds to swapping two columns in \(E^t\). 2. Multiplying a row in \(E\) by a nonzero scalar corresponds to multiplying a column in \(E^t\) by the same nonzero scalar. 3. Adding a multiple of one row to another row in \(E\) corresponds to adding the corresponding multiple of one column to another column in \(E^t\). Since all three types of elementary row operations are preserved in the transpose, if \(E\) is an elementary matrix, then \(E^t\) is also an elementary matrix, and vice versa.

Step by step solution

01

Swapping Two Rows

When swapping two rows in a matrix, we can find the transpose by swapping the corresponding columns in the transposed matrix. Since swapping two rows is an elementary row operation, and swapping two columns is the result of swapping rows in the transpose, this elementary operation on \(E\) is preserved in \(E^t\). For example, let \(E\) be the result of swapping row \(i\) and row \(j\) in identity matrix \(I\), then \(E^t\) can be obtained by swapping column \(i\) and column \(j\) in \(I^t\).
02

Multiplying a Row by a Nonzero Scalar

When multiplying a row by a nonzero scalar in a matrix, we can find the transpose by multiplying the corresponding column in the transposed matrix by the same scalar. Since multiplying a row by a nonzero scalar is an elementary row operation, and multiplying a column by a nonzero scalar in the transpose has the same effect, this elementary operation on \(E\) is preserved in \(E^t\). For example, let \(E\) be the result of multiplying row \(i\) in identity matrix \(I\) by scalar \(k\), then \(E^t\) can be obtained by multiplying column \(i\) in \(I^t\) by scalar \(k\).
03

Adding a Multiple of One Row to Another Row

When adding a multiple of one row to another row in a matrix, we can find the transpose by adding the corresponding multiple of one column to another column in the transposed matrix. Since adding a multiple of one row to another row is an elementary row operation, and adding a multiple of one column to another column in the transpose has the same effect, this elementary operation on \(E\) is preserved in \(E^t\). For example, let \(E\) be the result of adding a multiple (\(k\) times) of row \(i\) to row \(j\) in identity matrix \(I\), then \(E^t\) can be obtained by adding the same multiple (\(k\) times) of column \(i\) to column \(j\) in \(I^t\). By analyzing the behavior of the transpose with respect to the three types of elementary row operations, we can conclude that if \(E\) is an elementary matrix, then \(E^t\) is also an elementary matrix. The proof for the reverse direction follows the same logic since the transpose of \(E^t\) is \(E\).

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Key Concepts

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

Matrix Transpose
The transpose of a matrix is a fundamental concept in linear algebra that involves flipping a matrix over its diagonal. This means that the rows of the original matrix become the columns in the transposed matrix, and vice versa. The notation for the transpose of a matrix is given as \(A^{t}\), where \(A\) is the original matrix.

To transpose a matrix effectively, one should follow these steps:
  • Identify all the rows in the original matrix.
  • Turn those rows into columns to create the transposed matrix.
The transpose has diverse applications in mathematical computation and is crucial for operations such as solving systems of equations and finding determinants. It maintains several key mathematical properties:
  • The transpose of a transpose returns the original matrix: \((A^t)^t = A\).
  • The transpose of a sum of matrices is the sum of their transposes: \((A + B)^t = A^t + B^t\).
  • The transpose of a product reverses the order of multiplication: \((AB)^t = B^tA^t\).
Elementary Row Operations
Elementary row operations are fundamental procedures used to manipulate matrices and are essential in Gaussian elimination. These operations are used to simplify matrices, especially when solving systems of linear equations. They include:
  • Row swapping: This involves exchanging two rows in a matrix. It's a reversible operation and is often used to position a non-zero element in the pivot position.
  • Row multiplication: Every element in a row is multiplied by a non-zero scalar. This does not affect the solution set of a system of linear equations.
  • Row addition: A multiple of one row is added to another row to create zeros in specific positions, simplifying the matrix.


These operations can easily be carried out by using an identity matrix as the base. For example, transforming the identity matrix by row swapping results in an elementary matrix that can be used for subsequent matrix operations. Understanding these operations allows for easier manipulation of matrices in various operations and helps maintain crucial properties of matrices such as rank and determinant.
Linear Algebra Concepts
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It is foundational for understanding many aspects of mathematics and related fields such as engineering, computer science, and economics. A few key elements of linear algebra include:
  • Vectors and vector spaces: Vectors are objects that can be added together and multiplied by scalars to produce another vector. A vector space is a collection of vectors where vector addition and scalar multiplication are defined and satisfy eight axioms.
  • Matrices and determinants: A matrix is a rectangular array of numbers that can represent a system of linear equations. Determinants, computed from square matrices, are used in calculating matrix inverses and understanding matrix properties.
  • Eigenvalues and eigenvectors: These are pivotal in understanding the transformation properties of matrices and are widely used in areas like stability analysis and differential equations.

Understanding these concepts is critical in fields that solve real-world problems through mathematical modeling, including optimization, physics simulations, and financial analysis. Each concept interrelates and builds on the other, creating a cohesive framework for interpreting and solving complex problems in a structured way.

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Most popular questions from this chapter

Let \(A\) be an \(m \times n\) matrix with rank \(m\). Prove that there exists an \(n \times m\) matrix \(B\) such that \(A B=I_{m}\).

Let \(A\) be an \(m \times n\) matrix. Prove that if \(c\) is any nonzero scalar, then \(\operatorname{rank}(c A)=\operatorname{rank}(A)\).

Let \(\mathrm{T}, \mathrm{U}: \mathrm{V} \rightarrow \mathrm{W}\) be linear transformations. (a) Prove that \(R(T+U) \subseteq R(T)+R(U)\). (See the definition of the sum of subsets of a vector space on page 22.) (b) Prove that if \(\mathrm{W}\) is finite-dimensional, then \(\operatorname{rank}(\mathrm{T}+\mathrm{U}) \leq \operatorname{rank}(\mathrm{T})+\) \(\operatorname{rank}(U)\). (c) Deduce from (b) that \(\operatorname{rank}(A+B) \leq \operatorname{rank}(A)+\operatorname{rank}(B)\) for any \(m \times n\) matrices \(A\) and \(B\).

Let \(A\) be an \(m \times n\) matrix with rank \(m\) and \(B\) be an \(n \times p\) matrix with rank \(n\). Determine the rank of \(A B\). Justify your answer.

Let $$ A=\left(\begin{array}{rrrrr} 1 & 0 & -1 & 2 & 1 \\ -1 & 1 & 3 & -1 & 0 \\ -2 & 1 & 4 & -1 & 3 \\ 3 & -1 & -5 & 1 & -6 \end{array}\right) $$ (a) Find a \(5 \times 5\) matrix \(M\) with rank 2 such that \(A M=O\), where \(O\) is the \(4 \times 5\) zero matrix. (b) Suppose that \(B\) is a \(5 \times 5\) matrix such that \(A B=O\). Prove that \(\operatorname{rank}(B) \leq 2\).
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