Math Problem Statement

Given the following, find the reduced row echelon form of the augmented matrix [A | b]. (Enter your answer as one augmented matrix.) A =

2 4 4 6

2 4 6 8 and b =

10

14 RREF =

Solution

To find the reduced row echelon form (RREF) of the augmented matrix [Ab][A \mid b], we will apply the Gaussian elimination process, including row reductions to simplify the matrix to its RREF.

We are given the following system:

Matrix AA:

2 & 4 & 4 & 6 \\ 2 & 4 & 6 & 8 \end{pmatrix}$$ Column vector $$b$$: $$b = \begin{pmatrix} 10 \\ 14 \end{pmatrix}$$ ### Step 1: Form the augmented matrix $$[A \mid b]$$: $$[A \mid b] = \begin{pmatrix} 2 & 4 & 4 & 6 & 10 \\ 2 & 4 & 6 & 8 & 14 \end{pmatrix}$$ ### Step 2: Perform Gaussian elimination to reduce this to RREF. 1. **Row 1 normalization**: Make the leading coefficient of Row 1 (R1) equal to 1 by dividing the entire row by 2. $$R1 = \frac{1}{2} R1 \quad \Rightarrow \quad \begin{pmatrix} 1 & 2 & 2 & 3 & 5 \end{pmatrix}$$ Now, the augmented matrix becomes: $$\begin{pmatrix} 1 & 2 & 2 & 3 & 5 \\ 2 & 4 & 6 & 8 & 14 \end{pmatrix}$$ 2. **Row 2 elimination**: Eliminate the leading coefficient in Row 2 (R2) by subtracting 2 times Row 1 from Row 2. $$R2 = R2 - 2 \times R1 \quad \Rightarrow \quad \begin{pmatrix} 0 & 0 & 2 & 2 & 4 \end{pmatrix}$$ The matrix now becomes: $$\begin{pmatrix} 1 & 2 & 2 & 3 & 5 \\ 0 & 0 & 2 & 2 & 4 \end{pmatrix}$$ 3. **Row 2 normalization**: Make the leading coefficient of Row 2 (R2) equal to 1 by dividing the entire row by 2. $$R2 = \frac{1}{2} R2 \quad \Rightarrow \quad \begin{pmatrix} 0 & 0 & 1 & 1 & 2 \end{pmatrix}$$ The matrix now looks like: $$\begin{pmatrix} 1 & 2 & 2 & 3 & 5 \\ 0 & 0 & 1 & 1 & 2 \end{pmatrix}$$ 4. **Row 1 elimination**: Eliminate the third entry of Row 1 by subtracting 2 times Row 2 from Row 1. $$R1 = R1 - 2 \times R2 \quad \Rightarrow \quad \begin{pmatrix} 1 & 2 & 0 & 1 & 1 \end{pmatrix}$$ The augmented matrix becomes: $$\begin{pmatrix} 1 & 2 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 2 \end{pmatrix}$$ At this point, we have reduced the augmented matrix to its row echelon form. To continue simplifying: 5. **Row 1 normalization**: Eliminate the second entry in Row 1. Subtract 2 times the second column from Row 1. \[ R1 = R1 - 2R2 \quad \Rightarrow \quad \begin(turnover} stack

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Operations
Gaussian Elimination
Reduced Row Echelon Form (RREF)

Formulas

RREF involves row operations: Row scaling, Row swapping, Row addition/subtraction

Theorems

Gaussian Elimination
Properties of RREF

Suitable Grade Level

Undergraduate (Linear Algebra Course)