Convert System of Equations to Matrix Row Echelon Form

Math Problem Statement

muodosta tästä matriisi ja näytä välivaiheittain miten se muutetaa vaakarivi porrasmuotoon

Solution

To form a matrix from the given system of equations and then convert it to row echelon form (REF), let's start by identifying the coefficients in each equation.

Given System of Equations

x + y + 5z = 1 \\ -x - 2z = 3 \\ x + y + az = b \\ \end{cases}$$ ### Step 1: Write the System in Matrix Form The system can be represented in an augmented matrix form, where each row corresponds to an equation, and each column corresponds to the coefficients of $$x$$, $$y$$, and $$z$$, followed by the constant on the right side of each equation. $$\begin{pmatrix} 1 & 1 & 5 & | & 1 \\ -1 & 0 & -2 & | & 3 \\ 1 & 1 & a & | & b \\ \end{pmatrix}$$ ### Step 2: Row Operations to Achieve Row Echelon Form To convert the matrix to row echelon form, we aim to have leading 1s down the diagonal and zeros below each leading 1. #### Step 2.1: Make the First Element in the First Row a Leading 1 The first element is already 1 in the first row. So, the matrix remains: $$\begin{pmatrix} 1 & 1 & 5 & | & 1 \\ -1 & 0 & -2 & | & 3 \\ 1 & 1 & a & | & b \\ \end{pmatrix}$$ #### Step 2.2: Eliminate the x-Terms in Rows 2 and 3 Using Row 1 - **Row 2:** Add Row 1 to Row 2 to make the first element in Row 2 zero. $$R_2 = R_2 + R_1 \Rightarrow (0, 1, 3, |, 4)$$ - **Row 3:** Subtract Row 1 from Row 3 to make the first element in Row 3 zero. $$R_3 = R_3 - R_1 \Rightarrow (0, 0, a - 5, |, b - 1)$$ The matrix now becomes: $$\begin{pmatrix} 1 & 1 & 5 & | & 1 \\ 0 & 1 & 3 & | & 4 \\ 0 & 0 & a - 5 & | & b - 1 \\ \end{pmatrix}$$ #### Step 2.3: Ensure the Leading 1s and Check Row Echelon Form Now we have: 1. A leading 1 in the first column of the first row. 2. A leading 1 in the second column of the second row. 3. The third row has a leading term in the third column if $$a - 5 \neq 0$$. Thus, the matrix is now in row echelon form, assuming $$a - 5 \neq 0$$. If $$a - 5 = 0$$, special handling would be required as it would affect the row echelon structure. ### Final Matrix in Row Echelon Form $$\begin{pmatrix} 1 & 1 & 5 & | & 1 \\ 0 & 1 & 3 & | & 4 \\ 0 & 0 & a - 5 & | & b - 1 \\ \end{pmatrix}$$ This is the row echelon form of the matrix. Would you like further clarification or additional steps for specific cases? Here are some related questions to consider: 1. What happens if $$a = 5$$ in this system? 2. How do you interpret this system geometrically in terms of planes? 3. What additional steps are needed to find the values of $$x$$, $$y$$, and $$z$$ if $$a$$ and $$b$$ are given? 4. How does changing the values of $$a$$ and $$b$$ affect the solution? 5. Can you identify if the system has a unique solution, no solution, or infinitely many solutions? **Tip:** When performing row operations, always work systematically to avoid calculation errors, focusing on creating zeros in a step-by-step manner.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Row Echelon Form
System of Linear Equations

Formulas

Row operations to achieve row echelon form

Theorems

Matrix Row Echelon Form

Suitable Grade Level

Grades 10-12

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