Math Problem Statement

create the augmented matrix so the top corner is one firstl

Solution

To create an augmented matrix where the top-left corner (also called the leading coefficient) is 1, we typically refer to the process of row-reducing a matrix in linear algebra. Here's a general step-by-step guide to transforming the top-left corner of an augmented matrix into 1:

1. Set Up the Augmented Matrix:

The augmented matrix represents a system of linear equations, usually in the form:

a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22} & a_{23} & b_2 \\ a_{31} & a_{32} & a_{33} & b_3 \\ \end{array}$$ Where the left side represents the coefficients of the system and the right column (after the vertical bar) represents the constants on the right-hand side of the equations. ### 2. **Ensure the Top-Left Entry is 1**: If $$a_{11} \neq 1$$, you need to transform it into 1 by dividing the entire first row by $$a_{11}$$: $$R_1 \rightarrow \frac{R_1}{a_{11}}$$ This will give you the updated matrix: $$\begin{array}{ccc|c} 1 & \frac{a_{12}}{a_{11}} & \frac{a_{13}}{a_{11}} & \frac{b_1}{a_{11}} \\ a_{21} & a_{22} & a_{23} & b_2 \\ a_{31} & a_{32} & a_{33} & b_3 \\ \end{array}$$ ### Example: If the system of equations is: $$\begin{aligned} 2x + 3y + z &= 5 \\ x + y + z &= 3 \\ 3x - 2y + z &= 4 \\ \end{aligned}$$ The augmented matrix is: $$\begin{array}{ccc|c} 2 & 3 & 1 & 5 \\ 1 & 1 & 1 & 3 \\ 3 & -2 & 1 & 4 \\ \end{array}$$ Now, divide the first row by 2 to get a 1 in the top-left corner: $$R_1 \rightarrow \frac{R_1}{2}$$ $$\begin{array}{ccc|c} 1 & 1.5 & 0.5 & 2.5 \\ 1 & 1 & 1 & 3 \\ 3 & -2 & 1 & 4 \\ \end{array}$$ ### 3. **Proceed with Row Reduction**: Once the top-left corner is 1, you can proceed with further row operations to continue solving the system using Gaussian elimination or Gauss-Jordan elimination. Would you like to continue with the next steps, or any specific example? --- Here are some follow-up questions: 1. What is Gaussian elimination, and how does it help solve systems of equations? 2. Can you use row operations to make any augmented matrix into a reduced row echelon form? 3. How can you interpret the results of an augmented matrix in terms of solutions to the system? 4. What are the implications if a pivot in the augmented matrix is zero? 5. What is the difference between Gaussian elimination and Gauss-Jordan elimination? **Tip**: Always check your row operations for errors, as even a small mistake can affect the entire solution process.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Row Operations
Gaussian Elimination

Formulas

R1 → R1 / a11 (Row operation to make leading coefficient 1)

Theorems

Gaussian Elimination
Gauss-Jordan Elimination

Suitable Grade Level

Grades 10-12