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Chapter 2: Problem 70

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the augmented matrix corresponding to a system of three linear equations in three variables has a row of the form $\left[\begin{array}{lll|l}0 & 0 & 0 & a\end{array}\right]\(, where \)a$ is a nonzero number, then the system has no solution.

Short Answer

Expert verified
The statement is true. If the augmented matrix corresponding to a system of three linear equations in three variables has a row of the form \(\left[\begin{array}{lll|l}0 & 0 & 0 & a\end{array}\right]\), where \(a\) is a nonzero number, it represents the false equation \(0=a\). Such a false statement indicates that the system is inconsistent, and therefore, has no solution.

Step by step solution

01

Analyzing the given row

Since we have a row of the form \(\left[\begin{array}{lll|l}0 & 0 & 0 & a\end{array}\right]\), where \(a\) is a nonzero number, it means that we have a false statement. If we write down this row as one of the linear equations, it becomes: \[ 0x + 0y + 0z = a \] or this can be simplified as \[ 0 = a\] Given that \(a\) is a nonzero number, this equation is false as \(0\) cannot equal a nonzero number.
02

Conclusion

The statement is true. If the augmented matrix corresponding to a system of three linear equations in three variables has a row of the given form, it means that we have one of the linear equations giving us a false statement. This means that the system is inconsistent, and it has no solution.

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