Math Problem Statement
Row reduce the matrix to reduced echelon form. Identify the pivot positions in the final matrix and in the original matrix, and list the pivot columns. Start 3 By 4 Table 1st Row 1st Column 1 2nd Column 2 3rd Column 4 4st Column 3 2nd Row 1st Column 2 2nd Column 4 3rd Column 6 4st Column 2 3rd Row 1st Column 4 2nd Column 6 3rd Column 8 4st Column 0 EndTable
Question content area bottom Part 1 Row reduce the matrix to reduced echelon form and identify the pivot positions in the final matrix. The pivot positions are indicated by bold values. Choose the correct answer below. A.Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column negative 3 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column font size increased by 1 Bold 1 3rd Row 1st Column 0 2nd Column 0 3rd Column font size increased by 1 Bold 1 4st Column 0 EndTable Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column negative 3 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column font size increased by 1 Bold 1 3rd Row 1st Column 0 2nd Column 0 3rd Column font size increased by 1 Bold 1 4st Column 0 EndTable
B.Start 3 By 4 Table 1st Row 1st Column Bold font size increased by 1 1 2nd Column 0 3rd Column 0 4st Column negative 3 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column 2 3rd Row 1st Column 0 2nd Column 0 3rd Column Bold font size increased by 1 1 4st Column negative 2 EndTable Start 3 By 4 Table 1st Row 1st Column Bold font size increased by 1 1 2nd Column 0 3rd Column 0 4st Column negative 3 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column 2 3rd Row 1st Column 0 2nd Column 0 3rd Column Bold font size increased by 1 1 4st Column negative 2 EndTable
C.Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column 0 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column font size increased by 1 Bold 1 4st Column font size increased by 1 Bold 1 EndTable Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column 0 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column font size increased by 1 Bold 1 4st Column font size increased by 1 Bold 1 EndTable
D.Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column negative 1 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column negative 2 3rd Row 1st Column 0 2nd Column 0 3rd Column Bold font size increased by 1 1 4st Column 2 EndTable Start 3 By 4 Table 1st Row 1st Column font size increased by 1 Bold 1 2nd Column 0 3rd Column 0 4st Column negative 1 2nd Row 1st Column 0 2nd Column font size increased by 1 Bold 1 3rd Column 0 4st Column negative 2 3rd Row 1st Column 0 2nd Column 0 3rd Column Bold font size increased by 1 1 4st Column 2 EndTable
Solution
To row reduce the given matrix to reduced echelon form (REF) and identify the pivot positions, we proceed as follows:
Original Matrix:
1 & 2 & 4 & 3 \\ 2 & 4 & 6 & 2 \\ 4 & 6 & 8 & 0 \end{bmatrix}$$ ### **Step 1: Make the first pivot a 1.** The element in the first row and first column is already 1. ### **Step 2: Eliminate the entries below the pivot in column 1.** Using row operations: - $$R_2 \to R_2 - 2R_1$$ - $$R_3 \to R_3 - 4R_1$$ After these operations: $$\begin{bmatrix} 1 & 2 & 4 & 3 \\ 0 & 0 & -2 & -4 \\ 0 & -2 & -8 & -12 \end{bmatrix}$$ ### **Step 3: Normalize the second pivot.** The pivot in the second row is $$-2$$. Normalize: - $$R_2 \to -\frac{1}{2}R_2$$ This gives: $$\begin{bmatrix} 1 & 2 & 4 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & -2 & -8 & -12 \end{bmatrix}$$ ### **Step 4: Eliminate the third row's second column entry.** Using: - $$R_3 \to R_3 + 2R_2$$ Results: $$\begin{bmatrix} 1 & 2 & 4 & 3 \\ 0 & 0 & 1 & 2 \\ 0 & -2 & 0 & -8 \end{bmatrix}$$ ### **Step 5: Normalize the third pivot.** Normalize the pivot in $$R_3, C_3$$: - $$R_3 \to -\frac{1}{2}R_3$$ Final reduced row echelon form: $$\begin{bmatrix} 1 & 2 & 4 & 3 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \end{bmatrix}$$ ### **Identify Pivot Positions and Pivot Columns**: - Pivot positions: $$(1, 1), (2, 2), (3, 3)$$ - Pivot columns: $$C_1, C_2, C_3$$ ### **Correct Answer**: **A** The reduced echelon form matches option **A**. --- Do you want further clarifications or details? Here are some related questions for practice: 1. What are the row operations allowed in Gaussian elimination? 2. How do you verify the pivot positions in the original matrix? 3. Can a matrix with more columns than rows have a pivot in every column? 4. Explain why the reduced echelon form is unique. 5. What happens if a row of zeros appears during the row reduction process? **Tip**: Always double-check your row operations step by step to avoid computational errors!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Row Reduction
Reduced Row Echelon Form
Pivot Positions
Formulas
Row operations: swapping rows, scaling a row by a nonzero scalar, adding or subtracting multiples of rows
Definition of reduced row echelon form (REF)
Theorems
Fundamental theorem of row reduction
Uniqueness of reduced echelon form
Suitable Grade Level
College-Level Linear Algebra or Advanced High School Math