Problem 36 - 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 \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c\end{array}\right) \mathbf{A}\)

Step-by-Step Solution

Answer

The product matrix is \(\begin{pmatrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ c \cdot a_{31} & c \cdot a_{32} & c \cdot a_{33}\end{pmatrix}\).

Step 1: Identify the Elementary Matrix

The given matrix \(\mathbf{E}\) is \(\begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & c\end{pmatrix}\). This matrix represents an elementary matrix that scales the third row by \(c\). Such a matrix keeps the first and second rows unchanged and scales the third row of the matrix it multiplies.

Step 2: Multiply the Matrices

Multiply the elementary matrix \(\mathbf{E}\) by the arbitrary matrix \(\mathbf{A} = \begin{pmatrix}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}\end{pmatrix}\). The resulting matrix \(\mathbf{B} = \mathbf{E} \cdot \mathbf{A}\) will be \(\begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & c\end{pmatrix} \cdot \begin{pmatrix}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}\end{pmatrix}\).

Step 3: Compute Each Row of the Product

Compute each row of \(\mathbf{B}\) by executing matrix multiplication. The first row of \(\mathbf{B}\) is \(1 \cdot a_{11} + 0 \cdot a_{21} + 0 \cdot a_{31}= a_{11}\), \(1 \cdot a_{12} + 0 \cdot a_{22} + 0 \cdot a_{32} = a_{12}\), \(1 \cdot a_{13} + 0 \cdot a_{23} + 0 \cdot a_{33} = a_{13}\). The second row is \(0 \cdot a_{11} + 1 \cdot a_{21} + 0 \cdot a_{31}= a_{21}\), \(0 \cdot a_{12} + 1 \cdot a_{22} + 0 \cdot a_{32} = a_{22}\), \(0 \cdot a_{13} + 1 \cdot a_{23} + 0 \cdot a_{33} = a_{23}\). The third row is \(0 \cdot a_{11} + 0 \cdot a_{21} + c \cdot a_{31}= c \cdot a_{31}\), \(0 \cdot a_{12} + 0 \cdot a_{22} + c \cdot a_{32} = c \cdot a_{32}\), \(0 \cdot a_{13} + 0 \cdot a_{23} + c \cdot a_{33} = c \cdot a_{33}\).

Step 4: Final Matrix Product

Combine these calculations to form the matrix \(\mathbf{B}\). The resultant matrix \(\mathbf{B}\) is \(\begin{pmatrix}a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ c \cdot a_{31} & c \cdot a_{32} & c \cdot a_{33}\end{pmatrix}\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a process where we take two matrices and produce another matrix. Each entry in the resulting matrix is derived from the sum of products of corresponding entries in the rows of the first matrix and columns of the second.
This process requires that the number of columns in the first matrix is equal to the number of rows in the second. With this operation, we can transform datasets and solve various linear equations visually and algebraically.
In this exercise, we see how multiplying an elementary matrix with any given matrix \(\mathbf{A}\) results in a matrix where specified row operations have been performed on \(\mathbf{A}\). This highlights how multiplication is not limited to numbers but is also applied to matrices, which hold rows and columns of numbers in linear algebra.

Row Operations

In linear algebra, row operations are fundamental tools that modify matrices. They help solve matrix equations and analyze matrix structures. There are three types of row operations:

• Row swapping: Interchanging two rows in the matrix.
• Row multiplication: Multiplying all elements in a row by a non-zero scalar (a fancy word for just a number).
• Row addition: Adding a multiple of one row to another row.

Such operations are reversible, meaning they can be undone. They are used in algorithms like Gaussian elimination to find solutions to systems of linear equations. In the exercise, by multiplying \(\mathbf{A}\) with the elementary matrix \(\mathbf{E}\), we scale elements in the third row of \(\mathbf{A}\) by \(c\), which represents a row multiplication operation.

Scaling Matrix

A scaling matrix is a special kind of matrix that's used to transform other matrices by stretching or compressing their dimensions. A scaling matrix has values on its main diagonal (top-left to bottom-right), and these values dictate the factor by which certain parts of another matrix are scaled.
In our given problem, \(\mathbf{E}\) is a scaling matrix that leaves the first and second rows unchanged, while modifying the third row by the factor \(c\). When \(\mathbf{E}\) is multiplied with any general matrix \(\mathbf{A}\), it results in the third row entries being multiplied by \(c\) while other rows remain intact. This visualization illustrates how particular entries in a matrix can be targeted for transformation, emphasizing the importance of scaling matrices in topics like computer graphics and coordinate transformations.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces (also known as linear spaces) and linear transformations between them.
It's the foundation for many mathematical concepts and applications, like systems of linear equations, vector spaces, and matrices. This field is crucial for engineering, physics, computer science, economics, and more.
Working with matrices, as in our problem, showcases how linear algebra helps solve real-world problems by providing a framework to manipulate and analyze data. Understanding concepts like matrix multiplication, row operations, and scaling helps us in a wide range of applications, including algorithms in machine learning, solving differential equations, and optimization problems.