Question

Let \(A=\left[\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right]\) and \(C=\left[\begin{array}{rr}-1 & 1 \\ 2 & 1\end{array}\right]\). a. Find elementary matrices \(E_{1}\) and \(E_{2}\) such that \(C=E_{2} E_{1} A\). b. Show that there is \(n o\) elementary matrix \(E\) such that \(C=E A\)

Short answer

Elementary matrices are \(E_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\), \(E_2 = \begin{bmatrix} 1 & 0 \\ -1 & 2 \end{bmatrix}\). No single elementary matrix \(E\) can exist for \(C = EA\).

Step-by-step solution

Understanding the Problem

We have matrices \(A\) and \(C\) and need to find elementary matrices \(E_1\) and \(E_2\) such that \(C = E_2 E_1 A\). Then, check if a single elementary matrix \(E\) can transform \(A\) directly to \(C\) as \(C = EA\).

Determine the row operations for E1

Analyze the transformation required to change matrix \(A\) into an intermediate matrix. Start by making it look a bit closer to matrix \(C\). An initial operation can be swapping rows 1 and 2 of matrix \(A\). The corresponding elementary matrix for this operation is \(E_1 = \left[\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right]\).

Apply E1 to A

Calculate \(E_1 A\):\[E_1 A = \left[\begin{array}{cc} 0 & 1 \ 1 & 0 \end{array}\right] \left[\begin{array}{cc} 1 & 2 \ -1 & 1 \end{array}\right] = \left[\begin{array}{cc} -1 & 1 \ 1 & 2 \end{array}\right]\] This results in an intermediate matrix closer to \(C\).

Determine the row operations for E2

Next, make this intermediate matrix into \(C\). We need to change the second row \([1, 2]\) of the intermediate result to \([2, 1]\). This can be achieved by multiplying the second row by 2 and then subtracting the first row, row operation equivalent to "R2 -> 2R2 - R1." The elementary matrix for this operation is \(E_2 = \left[\begin{array}{cc} 1 & 0 \ -1 & 2 \end{array}\right]\).

Apply E2 to the intermediate result

Calculate \(E_2 (E_1 A)\) to verify it transforms the intermediate matrix to \(C\):\[E_2 \left(\left[\begin{array}{cc} -1 & 1 \ 1 & 2 \end{array}\right]\right) = \left[\begin{array}{cc} 1 & 0 \ -1 & 2 \end{array}\right] \left[\begin{array}{cc} -1 & 1 \ 1 & 2 \end{array}\right] = \left[\begin{array}{cc} -1 & 1 \ 2 & 1 \end{array}\right]\] This verifies that \(E_2 E_1 A = C\).

Consider a single elementary matrix transformation (Part b)

Realize that a single elementary matrix \(E\) corresponds to only one basic row operation (such as swap, scale, add multiple). Given that transforming \(A\) to \(C\) involves more than one operation, involving swapping and a row combination, no single elementary matrix \(E\) can suffice.

Key concepts

Matrix Transformation

Matrix transformation involves manipulating matrices using specific operations to achieve a desired form or result. In essence, when we say we transform matrix \(A\) into matrix \(C\), we are applying a sequence of processes or operations that change \(A\) step-by-step until it resembles \(C\). These operations can include row swaps, row scaling, or adding multiples of rows—all of which are integral in linear algebra.

• Elementary matrices come into play as they represent these row operations.
• Each row operation corresponds to multiplying \(A\) by an elementary matrix.

Understanding matrix transformations is crucial not only for solving linear equations but also for more advanced applications in computer graphics, data science, and differential equations.
Transforming a matrix is like solving a puzzle, where each move neatly alters your piece (matrix) to match the goal.

Row Operations

Row operations are fundamental tools in linear algebra that shape how matrices change. They allow for systematic changes to matrices by performing operations directly on the rows of the matrix. There are three primary types of row operations:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting the multiples of one row to another row.

These operations are very intuitive when you zoom in on elementary matrices. Each elementary matrix is born out of a specific row operation, serving as a basic building block.
For instance, if we swap the rows of \(A\), it will be represented by multiplying \(A\) by an elementary matrix correspondingly. This is crucial when simplifying systems of equations as well as performing Gaussian elimination to solve matrix equations.
Without row operations, the process of rectifying or modifying linear systems would be much more arduous.

Linear Algebra

Linear algebra is the branch of mathematics focusing on vector spaces and linear mappings between them. It’s the foundational language for a wide array of fields including engineering, physics, computer science, and more.

• Matrix calculations, vector spaces, and functions are central themes in linear algebra.
• Elementary matrices and row operations serve as the nuts and bolts.

Linear algebra is more than theory; it translates into real-world applications by explaining phenomena or solving complex systems efficiently.
Mastering it helps in understanding data transformations, optimizing processes, and modeling various scientific equations.
In brief, with linear algebra, we not only handle abstract problems but also connect theory with practical application.