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Chapter 1: Q26E (page 1)

Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.

Short Answer

Expert verified

No free variable lies in the system.

Step by step solution

01

Describe the given data

The coefficient matrix of a linear system of three equations in three variables has a pivot in each column.

02

Determine the existence of a solution

We know that a pivot is a nonzero real number. This implies that its echelon form of the augmented matrix has no row of the form

\(\left( {\begin{array}{*{20}{c}}0&0&0&b\end{array}} \right)\).

Here, \(b\) is any nonzero real number.

This ensures that there exists a solution.

03

Determine the uniqueness of a solution

Note that each row consists of a pivot. So, its row echelon form of the augmented matrix looks like an upper triangular matrix with nonzero diagonal entries. Therefore, the rank is 3. Also, it indicates that there is no free variable. Hence, there is no need for parameters.

04

Conclusion

Thus, the system has a unique solution.

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Most popular questions from this chapter

In Exercise 2, compute \(u + v\) and \(u - 2v\).

2. \(u = \left[ {\begin{array}{*{20}{c}}3\\2\end{array}} \right]\), \(v = \left[ {\begin{array}{*{20}{c}}2\\{ - 1}\end{array}} \right]\).

In (a) and (b), suppose the vectors are linearly independent. What can you say about the numbers \(a,....,f\) ? Justify your answers. (Hint: Use a theorem for (b).)

  1. \(\left( {\begin{aligned}{*{20}{c}}a\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\d\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\end{aligned}} \right)\)
  2. \(\left( {\begin{aligned}{*{20}{c}}a\\1\\0\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}b\\c\\1\\0\end{aligned}} \right),\left( {\begin{aligned}{*{20}{c}}d\\e\\f\\1\end{aligned}} \right)\)

Suppose the system below is consistent for all possible values of \(f\) and \(g\). What can you say about the coefficients \(c\) and \(d\)? Justify your answer.

27. \(\begin{array}{l}{x_1} + 3{x_2} = f\\c{x_1} + d{x_2} = g\end{array}\)

Question: If A is a non-zero matrix of the form,[a-bba] then the rank of A must be 2.

Explain why a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3},{{\mathop{\rm v}\nolimits} _4}} \right\}\) in \({\mathbb{R}^5}\) must be linearly independent when \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\) is linearly independent and \({{\mathop{\rm v}\nolimits} _4}\) is not in Span \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).
See all solutions

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