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Chapter 1: Q27E (page 1)

Suppose\(A\) is the \(3 \times 3\) zero matrix (with all zero entries). Describe the solution set of the equation \(Ax = 0\).

Short Answer

Expert verified

The solution set of the equation \(Ax = 0\) consists of all vectors in \({\mathbb{R}^3}\).

Step by step solution

01

Write matrix \(A\) as an augmented matrix

Write matrix \(A\) as an augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right].\)

\(\left[ {\begin{array}{*{20}{c}}0&0&0&0\\0&0&0&0\\0&0&0&0\end{array}} \right]\)

02

Determine the basic variable and free variables

The variables corresponding to pivot columns in the matrix are called basic variables.The other variable is called a free variable.

Here, \({x_1},{x_2}\), and \({x_3}\) are free variables.

03

Determine the solution set of the equation \(Ax = 0\)

The general solution of the system is:

\(\begin{aligned}{c}x = \left[ {\begin{aligned}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{aligned}} \right]\\ = {x_1}\left[ {\begin{aligned}{*{20}{c}}1\\0\\0\end{aligned}} \right] + {x_2}\left[ {\begin{aligned}{*{20}{c}}0\\1\\0\end{aligned}} \right] + {x_3}\left[ {\begin{aligned}{*{20}{c}}0\\0\\1\end{aligned}} \right]\end{aligned}\)

Each \(x\) in \({\mathbb{R}^3}\) satisfies \(Ax = 0\) if \(A\) is a \(3 \times 3\) zero matrix. Therefore, the solution set of the equation \(Ax = 0\) consists of all vectors in \({\mathbb{R}^3}\).

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Most popular questions from this chapter

Suppose \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2}} \right\}\) is a linearly independent set in \({\mathbb{R}^n}\). Show that \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm v}\nolimits} _2}} \right\}\) is also linearly independent.

In Exercises 13 and 14, determine if \(b\) is a linear combination of the vectors formed from the columns of the matrix \(A\).

13. \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&2\\0&3&5\\{ - 2}&8&{ - 4}\end{array}} \right],{\mathop{\rm b}\nolimits} = \left[ {\begin{array}{*{20}{c}}3\\{ - 7}\\{ - 3}\end{array}} \right]\)

Let \({{\mathop{\rm a}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\4\\{ - 2}\end{array}} \right],{{\mathop{\rm a}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 3}\\7\end{array}} \right],\) and \({\rm{b = }}\left[ {\begin{array}{*{20}{c}}4\\1\\h\end{array}} \right]\). For what values(s) of \(h\) is \({\mathop{\rm b}\nolimits} \) in the plane spanned by \({{\mathop{\rm a}\nolimits} _1}\) and \({{\mathop{\rm a}\nolimits} _2}\)?

In Exercises 31, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

31. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\4&{ - 1}&3&{ - 6}\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&1&0\\0&5&{ - 2}&8\\0&7&{ - 1}&{ - 6}\end{array}} \right]\)

Question: Determine whether the statements that follow are true or false, and justify your answer.

14: rank.|111123136|=3
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