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Chapter 6: Problem 34

Describe what happens when Gaussian elimination is used to solve a system with dependent equations.

Short Answer

Expert verified
When applying Gaussian elimination to a system with dependent equations (infinite solutions), the result is a matrix with a row (or rows) of all zeros aside from the constant on the right side of the equation. This indicates that the system has dependent equations with an infinite number of solutions.

Step by step solution

01

Initial System

Consider a potential system with dependent equations: \(x + y + z = 4\), \(2x + 2y + 2z = 8\), and \(3x + 3y + 3z = 12\).
02

Apply Gaussian Elimination

Firstly, process of Gaussian elimination is applied to this system. This involves swapping equations around, multiplying entire equations by scalars, and adding equations to others in order to create a matrix with a clear diagonal from top-left to bottom-right and zeroes below this diagonal. For the system of dependent equations, we can simplify by dividing the second equation by 2 and third equation by 3 to obtain: \(x + y + z = 4\), \(x + y + z = 4\), and \(x + y + z = 4\).
03

Identify Dependent System

These equations are all the same, which illustrates how every solution to one equation will also be a solution to the others. This shows that our system of equations is dependent.
04

Final Matrix Form

After applying Gaussian elimination, you'll find that you end up with a row of zeroes in your matrix. In our case, After subtracting equation 2 from equation 1 or equation 3 from equation 1 or equation 2, we end up with the row of zeros in the matrix. Hence, The resultant matrix might look something like this: \(1x + 1y + 1z = 4\), \(0x + 0y + 0z = 0\), and \(0x + 0y + 0z = 0\).

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Most popular questions from this chapter

The process of solving a linear system in three variables using Cramer's rule can involve tedious computation. Is there a way of speeding up this process, perhaps using Cramer's rule to find the value for only one of the variables? Describe how this process might work, presenting a specific example with your description. Remember that your goal is still to find the value for each variable in the system.

In Exercises \(5-8,\) find values for the variables so that the matrices in each exercise are equal. $$ \left[\begin{array}{rr} x & 2 y \\ z & 9 \end{array}\right]=\left[\begin{array}{rr} 4 & 12 \\ 3 & 9 \end{array}\right] $$

Use Cramer's rule to solve each system. $$ \begin{aligned}&x+y+z=4\\\&x-2 y+z=7\\\&x+3 y+2 z=4\end{aligned} $$

Evaluate each determinant. $$ \left|\begin{array}{rrr}2 & -4 & 2 \\\\-1 & 0 & 5 \\\3 & 0 & 4\end{array}\right| $$

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A-C $$
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