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Chapter 1: Q16E (page 1)

In Exercises 15 and 16, fill in the missing enteries of the matrix, assuming that the equation holds for all values of the variables

\(\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\)

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\{ - 2}&1\\1&0\end{array}} \right]\)

Step by step solution

01

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the first row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}\end{array}} \right]\).

02

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the second row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}{ - 2}&1\end{array}} \right]\).

03

Compare the rows of the matrix

Compare both sides of the equation \[\left[ {\begin{array}{*{20}{c}}?&?\\?&?\\?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{x_1} - {x_2}}\\{ - 2{x_1} + {x_2}}\\{{x_1}}\end{array}} \right]\] to get the third row of the matrix with unknown elements as \(\left[ {\begin{array}{*{20}{c}}1&0\end{array}} \right]\).

So, the matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}\\{ - 2}&1\\1&0\end{array}} \right]\).

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Most popular questions from this chapter

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

Find an equation involving \(g,\,h,\)and \(k\) that makes this augmented matrix correspond to a consistent system:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&g\\0&3&{ - 5}&h\\{ - 2}&5&{ - 9}&k\end{array}} \right]\)


Consider two vectors v鈫�1 andv鈫�2in R3 that are not parallel.

Which vectors inlocalid="1668167992227" 鈩�3are linear combinations ofv鈫�1andv鈫�2? Describe the set of these vectors geometrically. Include a sketch in your answer.

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

In Exercise 23 and 24, make each statement True or False. Justify each answer.

23.

a. Another notation for the vector \(\left[ {\begin{array}{*{20}{c}}{ - 4}\\3\end{array}} \right]\) is \(\left[ {\begin{array}{*{20}{c}}{ - 4}&3\end{array}} \right]\).

b. The points in the plane corresponding to \(\left[ {\begin{array}{*{20}{c}}{ - 2}\\5\end{array}} \right]\) and \(\left[ {\begin{array}{*{20}{c}}{ - 5}\\2\end{array}} \right]\) lie on a line through the origin.

c. An example of a linear combination of vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) is the vector \(\frac{1}{2}{{\mathop{\rm v}\nolimits} _1}\).

d. The solution set of the linear system whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\) is the same as the solution set of the equation\({{\mathop{\rm x}\nolimits} _1}{a_1} + {x_2}{a_2} + {x_3}{a_3} = b\).

e. The set Span \(\left\{ {u,v} \right\}\) is always visualized as a plane through the origin.
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