Math Problem Statement
Hope has rewritten a system of equations as the following matrix: 2 –1 –3 1 2 –4
Now, Hope wants to convert this matrix into row echelon form.
Which of the following can she do first?
Select THREE correct answers. A.She can switch rows #1 and #2. B.She can multiply row #2 by −2. C.She can add −1 to each element of row #2. D.She can subtract each element of row #2 from its corresponding element in row #1. E.She can switch the last element of row #1 with its corresponding element in row #2.
Solution
Hope has the following matrix:
2 & -1 & -3 \\ 1 & 2 & -4 \end{pmatrix}$$ To convert this matrix into **row echelon form**, the goal is to get leading 1's and zeros below the leading entries (pivots). The following steps are possible in this process: ### Let’s examine the options one by one: 1. **Option A: She can switch rows #1 and #2.** - **Correct**. Row switching is a valid elementary row operation in Gaussian elimination. It can be helpful to get a leading 1 in the first row. By switching rows, Hope will place a 1 (from the second row) at the top. 2. **Option B: She can multiply row #2 by −2.** - **Correct**. Multiplying a row by a non-zero scalar is a valid elementary row operation. This can help create a more convenient form, such as getting rid of fractions or simplifying a row. 3. **Option C: She can add −1 to each element of row #2.** - **Incorrect**. Adding the same constant to every element in a row is **not** a valid elementary row operation. Instead, Hope can add or subtract rows or multiples of rows from each other. 4. **Option D: She can subtract each element of row #2 from its corresponding element in row #1.** - **Correct**. This operation is valid because subtracting one row from another is a standard row operation. It can help eliminate terms to create zeros. 5. **Option E: She can switch the last element of row #1 with its corresponding element in row #2.** - **Incorrect**. Swapping individual elements within a matrix is **not** a valid elementary row operation in Gaussian elimination. ### The correct answers are: - **A.** She can switch rows #1 and #2. - **B.** She can multiply row #2 by −2. - **D.** She can subtract each element of row #2 from its corresponding element in row #1. Would you like further details on how row echelon form is achieved or any other aspect? --- Here are 5 related questions for further practice: 1. How can Hope use row operations to simplify a 3x3 matrix? 2. What are the steps for solving a system of equations using Gaussian elimination? 3. Can Hope use a row operation to create a leading 1 in any row? 4. What is the difference between row echelon form and reduced row echelon form? 5. How does Hope identify when her matrix is in row echelon form? **Tip:** Always aim to get a leading 1 in each row, starting from the top-left, and create zeros below it as you work through the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Gaussian Elimination
Row Echelon Form
Elementary Row Operations
Formulas
-
Theorems
Gaussian Elimination Theorem
Suitable Grade Level
Grades 10-12