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It is shown early on in most linear algebra courses that every invertible matrix can be written as a product of elementary matrices (or, equivalently, that every invertible matrix can be reduced to the identity matrix by a finite number of row reduction steps). Show that every \(2 \times 2\) matrix of determinant 1 is the product of three elementary matrices. \((2 \times 2\) elementary matrices are matrices of types $$ \left(\begin{array}{ll} 1 & x \\ 0 & 1 \end{array}\right), \quad\left(\begin{array}{ll} 1 & 0 \\ x & 1 \end{array}\right), \quad\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), \quad\left(\begin{array}{ll} y & 0 \\ 0 & 1 \end{array}\right), \quad\left(\begin{array}{ll} 1 & 0 \\ 0 & y \end{array}\right), $$ where \(y \neq 0\) and \(x\) are arbitrary. The standard row reduction of \(\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\) would ordinarily use four row reduction steps.)

Step by step solution

01

Understand the Problem

Given a 2x2 matrix with determinant 1, we need to express it as a product of three elementary matrices.

02

Set Up the Target 2x2 Matrix

Consider a general 2x2 matrix: The determinant condition gives us: .

03

Identify Elementary Matrix Types

Based on the problem, we can use the following types of elementary matrices: 1. , which represents a row operation: 2. , which represents a row operation: 3. , which represents a row operation: 4. or , scaling operations for rows 1 or 2 respectively (where ).

04

Express the Determinant 1 Matrix

Since our target matrix has determinant 1, we need to represent it using three of the given types of elementary matrices:1. Start with where x and y are variables.2. Choose appropriate variables and combine.3. Verify the product results in a determinant of 1.

05

Verification

Multiply the chosen elementary matrices step-by-step to confirm that the resulting matrix has the determinant equal to 1.

06

Example Construction

For instance: 1. Use to introduce the off-diagonal term.2. Select to scale a row appropriately.3. Choose to adjust the other row.Multiply these together and check the product matrix matches the target matrix.

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Key Concepts

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

Determinant 1 Matrices

In linear algebra, a matrix's determinant provides important information about the matrix, such as whether it is invertible. For a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is calculated as \(ad - bc\). If the determinant is 1, this matrix is special because it is invertible and its inverse also has a determinant of 1. Any 2x2 matrix with determinant 1 can be written as a product of elementary matrices. This property forms the basis for various matrix operations like row reduction and matrix factorization. Elementary matrices corresponding to simple row operations can be used to transform matrices while preserving their structure, forming a critical concept in matrix theory.

Matrix Factorization

Matrix factorization is the process of breaking down a matrix into products of simpler matrices. For invertible 2x2 matrices with a determinant of 1, we can express these matrices as products of three elementary matrices. Elementary matrices are derived from elementary row operations, which include swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another. Factorizing a matrix into elementary matrices helps in simplifying matrix equations and solving systems of linear equations. For example, a matrix with a determinant of 1 can be broken down into the product of elementary matrices as follows:

• Start with matrix \(\begin{pmatrix} 1 & x \ 0 & 1 \end{pmatrix}\) to introduce an off-diagonal element.
• Use matrix \(\begin{pmatrix} y & 0 \ 0 & 1 \end{pmatrix}\) to scale the first row.
• Finally, choose matrix \(\begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}\) to swap rows if necessary.

This approach helps breakdown the complexity of matrix operations into simpler transformations.

Row Reduction

Row reduction is a method used to simplify a matrix into a more manageable form, usually the identity matrix. It is also known as Gaussian elimination. For 2x2 matrices with a determinant of 1, row reduction involves performing specific row operations to convert the given matrix into the identity matrix \(\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\). The row operations used in this process can be represented by elementary matrices. Performing row reduction with elementary matrices involves:

• Adding or subtracting multiples of rows to other rows.
• Swapping the positions of two rows.
• Scaling rows by non-zero constants.

By applying these operations, we step-by-step convert the original matrix into the identity matrix, proving that a matrix can indeed be simplified using elementary row operations. This method is critical in solving systems of linear equations and finding matrix inverses.