Math Problem Statement
Determine whether it is possible to write the matrix M = 1 0 1 −2 1 −2 0 2 1 as a product of elementary matrices. If you claim that it is possible to write M as a product of elementary matrices, then find elementary matrices whose product is M. If you claim that it is not possible to write M as a product of elementary matrices, then justify your claim.
Solution
Determining Whether is a Product of Elementary Matrices
Given Matrix:
\begin{bmatrix} 1 & 0 & 1 \\ -2 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix}.$$ A matrix $$M$$ can be written as a product of elementary matrices if $$M$$ is invertible. Invertibility of $$M$$ is determined by checking whether the determinant $$\det(M) \neq 0$$. --- ### Step 1: Compute $$\det(M)$$ The determinant of a $$3 \times 3$$ matrix is calculated using: $$\det(M) = a(ei − fh) − b(di − fg) + c(dh − eg),$$ where the matrix elements are: $$M = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}.$$ Here: $$a = 1, \, b = 0, \, c = 1, \, d = -2, \, e = 1, \, f = -2, \, g = 0, \, h = 2, \, i = 1.$$ Substituting into the formula: $$\det(M) = 1((1)(1) - (-2)(2)) - 0((-2)(1) - (-2)(0)) + 1((-2)(2) - (1)(0)).$$ Simplify: $$\det(M) = 1(1 + 4) + 1(-4).$$ $$\det(M) = 5 - 4 = 1.$$ --- ### Step 2: Interpret the Determinant Since $$\det(M) = 1 \neq 0$$, the matrix $$M$$ is invertible. Therefore, $$M$$ **can be expressed as a product of elementary matrices**. --- ### Step 3: Find Elementary Matrices To write $$M$$ as a product of elementary matrices, perform Gaussian elimination on $$M$$ to reduce it to the identity matrix $$I$$, while keeping track of the row operations. #### Initial Matrix: $$M = \begin{bmatrix} 1 & 0 & 1 \\ -2 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix}.$$ 1. **Row Operation 1:** $$R_2 \to R_2 + 2R_1$$ $$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 2 & 1 \end{bmatrix}.$$ The corresponding elementary matrix is: $$E_1 = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ 2. **Row Operation 2:** $$R_3 \to R_3 - 2R_2$$ $$\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ The corresponding elementary matrix is: $$E_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -2 & 1 \end{bmatrix}.$$ 3. **Row Operation 3:** $$R_1 \to R_1 - R_3$$ $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I.$$ The corresponding elementary matrix is: $$E_3 = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ --- ### Final Result The matrix $$M$$ can be expressed as: $$M = E_1^{-1} E_2^{-1} E_3^{-1},$$ where: $$E_1 = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \, E_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -2 & 1 \end{bmatrix}, \, E_3 = \begin{bmatrix} 1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$$ --- ### Summary - **It is possible** to write $$M$$ as a product of elementary matrices. - The matrices $$E_1, E_2, E_3$$ and their inverses are explicitly determined. Would you like to see detailed steps for inverse matrices or further explanation? --- ### Related Questions 1. What properties of determinants ensure invertibility of a matrix? 2. Can every invertible matrix be written as a product of elementary matrices? 3. How does Gaussian elimination relate to elementary matrices? 4. What are the common types of elementary matrices, and how are they used? 5. How do you compute the inverse of an elementary matrix? --- ### Tip When performing row operations, always verify each step by multiplying matrices to check accuracy!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Invertibility
Elementary Matrices
Gaussian Elimination
Formulas
Determinant formula for a 3x3 matrix: det(M) = a(ei − fh) − b(di − fg) + c(dh − eg)
Properties of determinant: det(M) ≠ 0 implies M is invertible
Theorems
Invertibility theorem: A matrix is invertible if and only if its determinant is non-zero
Elementary Matrix Theorem: Every invertible matrix can be written as a product of elementary matrices
Suitable Grade Level
Undergraduate Level (Linear Algebra Course)
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