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Chapter 6: Problem 38

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

Short Answer

Expert verified
When Gaussian elimination is used to solve a system with dependent equations, during the process of row reduction, a zero row is obtained in the resultant augmented matrix. This indicates the presence of an infinite number of solutions, characteristic of dependent systems.

Step by step solution

01

Understand dependent systems

Dependent systems of equations have infinitely many solutions. For such systems, there are either the same equations or an equation that is a multiple of another, which makes them dependent.
02

Performing Gaussian elimination

When you start performing Gaussian elimination on a dependent system, you would get a row of zeros in your augmented matrix during row reduction.
03

Interpretation

The row of zeros is indicating that the equations are identical or one is a multiple of the other, meaning they are dependent. Hence, there is an infinite number of solutions, which is what we expected from a dependent system.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can speed up the tedious computations required by Cramer's Rule by using the value of \(D\) to determine the value of \(D_{x^{*}}\)

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{rrr}1 & -1 & 4 \\\4 & -1 & 3 \\\2 & 0 & -2\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 1 & 0 \\\1 & 2 & 4 \\\1 & -1 & 3\end{array}\right]$$

The figure shows the letter \(L\) in a rectangular coordinate system. (GRAPH CANNOT COPY) The figure can be represented by the matrix $$B=\left[\begin{array}{llllll}0 & 3 & 3 & 1 & 1 & 0 \\\0 & 0 & 1 & 1 & 5 & 5\end{array}\right]$$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The \(L\) is completed by connecting the last point in the matrix, \((0,5),\) to the starting point, \((0,0) .\) Use these ideas to solve Exercises \(53-60 .\) a. If \(A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right],\) find \(A B\) b. Graph the object represented by matrix \(A B\). What effect does the matrix multiplication have on the letter \(L\) represented by matrix \(B ?\)

Describe how to multiply matrices.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When using Cramer's Rule to solve a linear system, the number of determinants that I set up and evaluate is the same as the number of variables in the system.
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