Problem 35 - Advanced Engineering Mathematics

Question

If a matrix \(\mathbf{A}\) is premultiplied by an elementary matrix \(\mathbf{E}\), the product EA will be that matrix obtained from A by performing the elementary row operation symbolized by \(\mathbf{E}\). Compute the given product for an arbitrary \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right) \mathbf{A}\)

Step-by-Step Solution

Answer

The operation swaps the first two rows of matrix \( \mathbf{A} \).

Step 1: Understand the Operation of the Elementary Matrix

The given elementary matrix \( \mathbf{E} = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \) is designed to swap the first and second rows of any 3x3 matrix it is multiplied by, like \( \mathbf{A} \).

Step 2: Represent the Arbitrary Matrix A

Let \( \mathbf{A} \) be defined as \( \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix} \). This general form helps us calculate the specific row operations when \( \mathbf{E} \) is applied.

Step 3: Perform the Multiplication EA

Multiply \( \mathbf{E} \) by \( \mathbf{A} \):\[\begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix} = \begin{pmatrix} a_{21} & a_{22} & a_{23} \ a_{11} & a_{12} & a_{13} \ a_{31} & a_{32} & a_{33} \end{pmatrix} \]This operation swaps the first two rows of \( \mathbf{A} \), as visualized in the result.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a third matrix. Each element in the result matrix is the dot product of a row in the first matrix and a column in the second matrix. This operation is not commutative, meaning that multiplying matrix \( \mathbf{A} \) by \( \mathbf{B} \) does not yield the same result as multiplying \( \mathbf{B} \) by \( \mathbf{A} \). These products will, in most cases, be entirely different matrices.
To find the product of two matrices, you need to ensure that the number of columns in the first matrix is equal to the number of rows in the second matrix. This condition is necessary because otherwise, the elements cannot be paired correctly to calculate the dot products. Matrix multiplication is used widely across various fields including physics, computer science, and engineering for solving systems of equations, transforming geometric data, and many other applications.
For example, in the given solution, the elementary matrix \( \mathbf{E} \) is multiplied by a matrix \( \mathbf{A} \). Each element of the product matrix is formed by performing the dot product of corresponding row and column elements from these matrices.

Row Operations

Row operations are methods used to manipulate the rows of a matrix. These operations include row swapping, scaling rows by a nonzero scalar, and adding or subtracting multiples of rows. They are particularly helpful in solving systems of linear equations, as they maintain the equality's integrity without altering the solutions.
Elementary matrices are matrices that perform these row operations. The given matrix \( \mathbf{E} \) is an example of an elementary matrix intended for row swapping. Specifically, it will swap the first and second rows in any matrix it multiplies. This can be visualized by looking at how multiplying by \( \mathbf{E} \) transforms matrix \( \mathbf{A} \), where the result switches rows one and two.

• Row Swapping: Changes the position of rows.
• Row Scaling: Multiplies elements in a row by a constant.
• Row Addition: Adds elements from one row to another, often to help eliminate variables.

Mastering row operations allows you to effectively reduce matrices to simpler forms, like the row echelon form or reduced row echelon form, which greatly aids in solving equations.

3x3 Matrices

3x3 matrices are matrices with three rows and three columns, represented as a grid of nine elements. These matrices are powerful tools in mathematics and are often used for transformations involving two-dimensional or three-dimensional space.
The general form of a 3x3 matrix \( \mathbf{A} \) is given by:\[\mathbf{A} = \begin{pmatrix}a_{11} & a_{12} & a_{13} \a_{21} & a_{22} & a_{23} \a_{31} & a_{32} & a_{33}\end{pmatrix}\]
Such matrices play crucial roles in fields like physics and computer graphics for representing linear transformations. They can describe rotations, reflections, scaling, and translations in 3D space.
When dealing with 3x3 matrices, remember that certain properties like the determinant and trace become important. The determinant of a matrix helps determine properties such as invertibility and volume scaling factor under transformation. Working through operations like matrix multiplication with 3x3 matrices provides a solid foundation for handling more complex matrices and higher-dimensional transformations.