Math Problem Statement
Find a sequence of elementary matrices that can be used to write the given matrix in row-reduced form.
Solution
This problem asks to find a sequence of elementary matrices that can be used to express the given matrix in row-reduced form.
The matrix in question is:
0 & 1 & 9 \\ 5 & 15 & -5 \end{pmatrix}$$ Elementary matrices correspond to the row operations needed to convert this matrix into its row echelon form. Here are the steps to solve the system: ### Step 1: Swap rows (R1 ↔ R2) The first elementary operation needed is to swap rows to bring a non-zero entry to the top-left position. This yields: $$\begin{pmatrix} 5 & 15 & -5 \\ 0 & 1 & 9 \end{pmatrix}$$ The elementary matrix that corresponds to this row swap is: $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$ ### Step 2: Scale the first row (R1 → $$\frac{1}{5} \cdot R1$$) Now, divide the first row by 5 to make the leading entry a 1: $$\begin{pmatrix} 1 & 3 & -1 \\ 0 & 1 & 9 \end{pmatrix}$$ The elementary matrix for this operation is: $$\begin{pmatrix} \frac{1}{5} & 0 \\ 0 & 1 \end{pmatrix}$$ ### Step 3: No further row operations needed At this stage, the matrix is already in row-reduced form with leading 1's and zeroes below them. ### Correct Answer Based on the provided answer choices, the correct elementary matrices will correspond to: $$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} \frac{1}{5} & 0 \\ 0 & 1 \end{pmatrix}$$ This matches the second answer option in the image. --- Would you like more details on elementary matrices or any other part of the solution? Here are some related questions you can try: 1. What are the different types of elementary row operations? 2. How do you perform Gaussian elimination using elementary matrices? 3. Can you explain how to invert an elementary matrix? 4. Why are elementary matrices always invertible? 5. How can I use row operations to find the determinant of a matrix? **Tip:** When dealing with matrices, remember that each elementary matrix corresponds to a specific row operation that modifies the identity matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Elementary Matrices
Row Operations
Matrix Row Reduction
Formulas
Matrix row swap
Row scaling operation
Theorems
Gaussian Elimination
Elementary Matrix Theorem
Suitable Grade Level
College level or advanced high school