Math Problem Statement

An invertible matrix A can be written as a product of elementary matrices as follows: A = (2 0 0)(1 0 0)(0 1 0)(1 1 0)(1 1 0). Find the determinant of A.

Solution

Let's analyze the problem.

Problem Summary

The given matrix AA is written as a product of elementary matrices. The question asks us to determine the determinant of AA.

Key Points:

  1. If A=E1E2E3⋯EnA = E_1 E_2 E_3 \cdots E_n (where EiE_i are elementary matrices), then: det⁡(A)=det⁡(E1)×det⁡(E2)×⋯×det⁡(En).\det(A) = \det(E_1) \times \det(E_2) \times \cdots \times \det(E_n).
  2. We need to compute the determinant of each elementary matrix involved and then multiply them.

Step-by-Step Determinant Calculation

We are given:

\begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$$ Let’s compute the determinant of each matrix: 1. **First matrix**: $$\det\left(\begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right) = 2.$$ 2. **Second matrix**: $$\det\left(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & 0 & 1 \end{pmatrix}\right) = 1.$$ (This is a row operation of type 3, adding a multiple of one row to another, which doesn’t change the determinant.) 3. **Third matrix**: $$\det\left(\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right) = -1.$$ (This is a row swap, which introduces a factor of -1.) 4. **Fourth matrix**: $$\det\left(\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right) = 1.$$ 5. **Fifth matrix**: $$\det\left(\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right) = 1.$$ --- ### Final Determinant Calculation Now, multiply all the determinants: $$\det(A) = 2 \times 1 \times (-1) \times 1 \times 1 = -2.$$ --- ### Answer The determinant of $$A$$ is: $$\boxed{-2}.$$ --- Do you have any further questions or need clarification on any step? --- ### Related Questions 1. What are the different types of elementary matrices? 2. How does a row swap affect the determinant? 3. What is the determinant of a block diagonal matrix? 4. How do row operations relate to elementary matrices? 5. Why do some row operations leave the determinant unchanged? --- ### Tip When multiplying matrices for determinants, keep an eye on row swaps, as they introduce a factor of -1 into the product.

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Math Problem Analysis

Mathematical Concepts

Matrix Determinants
Elementary Matrices
Matrix Multiplication

Formulas

det(A) = det(E1) × det(E2) × ... × det(En)
Determinant of diagonal matrix: product of diagonal elements

Theorems

Properties of Determinants
Determinant of Elementary Matrices

Suitable Grade Level

Undergraduate Linear Algebra

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