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Chapter 4: Problem 209

Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: $$ \left[\begin{array}{rrr|r} 1 & -2 & 3 & -4 \\ 3 & -1 & -2 & 5 \\ 2 & -3 & -4 & -1 \end{array}\right] $$

Short Answer

Expert verified
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Step by step solution

01

Identify the key leading entry in Row 1

The key leading entry in Row 1 is the first element, which is 1.
02

Make the first entry in Row 2 zero

To make the first entry in Row 2 zero, subtract 3 times Row 1 from Row 2:
03

Calculate Row 2

Row 2 - 3(R1):
04

New values of Row 2

Calculate the values of the new Row 2:
05

Make the first entry in Row 3 zero

Calculate the coefficients and term for the new values.
06

Calculate Remaining Rows and check final augmented matrix form _Insert

_Insert Detailed row operations

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Row Operations
Row operations are crucial in Gaussian Elimination, which is a method used to solve systems of linear equations. These operations are applied to matrices in order to simplify them and make the solving process straightforward. Row operations include:
  • Swapping two rows
  • Multiplying a row by a non-zero scalar
  • Adding or subtracting one row to/from another row
For example, in our exercise, we need to make the first entry in both Row 2 and Row 3 zero. This requires adding or subtracting multiples of Row 1 to/from Rows 2 and 3.
Augmented Matrix
An augmented matrix is used to represent a system of linear equations. It includes the coefficients and constants from the equations. The augmented matrix from our given system of equations is: is displayed as: is displayed as: example_children1

This augmented matrix divides the coefficients and constants with a vertical line, making it easier to see the full system as a single entity.
Elementary Row Operations
Understanding elementary row operations is essential because they are the foundation of Gaussian Elimination. Here are the three types in more detail:
  • Swapping two rows: You can exchange any two rows to achieve a more manageable matrix form.
  • Multiplying a row by a non-zero scalar: This is useful when you want to normalize a row to simplify further operations.
  • Adding or subtracting multiples of one row to/from another row: This helps in eliminating variables to simplify the matrix into its row-echelon form.
For example, in the exercise, subtracting multiples of Row 1 from Rows 2 and 3 are elementary row operations employed to achieve zeros in the leading entries.
Leading Entry
The leading entry is the first non-zero number from the left in a row of a matrix. It's also termed as the pivot. In our given matrix: $$\begin{matrix} 1 & -2 & 3 & -4 \ 3 & -1 & -2 & 5 \ 2 & -3 & -4 & -1 \ \right) The leading entry of Row 1 is 1, which guides the row operations needed. We used this leading entry to make the first entries in both Row 2 and Row 3 zero. These operations drive the whole Gaussian Elimination forward by strategically simplifying the matrix.

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Most popular questions from this chapter

Solve each system of equations using Cramer's Rule. $$ \left\\{\begin{array}{l} 4 x+3 y=2 \\ 20 x+15 y=5 \end{array}\right. $$

Solve each system of equations using Cramer's Rule. $$ \left\\{\begin{array}{l} 4 x-3 y+z=7 \\ 2 x-5 y-4 z=3 \\ 3 x-2 y-2 z=-7 \end{array}\right. $$

Solve each system of equations using Cramer's Rule. $$ \left\\{\begin{array}{l} 2 x+3 y+z=12 \\ x+y+z=9 \\ 3 x+4 y+2 z=20 \end{array}\right. $$

Write the system of equations that corresponds to the augmented matrix. $$ \left[\begin{array}{ll|l} 2 & -4 & -2 \\ 3 & -3 & -1 \end{array}\right] $$

Solve each system by graphing. $$ \left\\{\begin{array}{l} y \leq-\frac{1}{2} x+3 \\ y<1 \end{array}\right. $$
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