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Chapter 1: Q15E (page 1)

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

\(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\)

Short Answer

Expert verified

\(\left[ {\begin{array}{*{20}{c}}3&0&{ - 2}\\4&0&0\\1&{ - 1}&1\end{array}} \right]\)

Step by step solution

01

Comparison of rows of the matrix

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the first row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}3&0&{ - 2}\end{array}} \right]\).

02

Comparison of rows of the matrix

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the second row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}4&0&0\end{array}} \right]\).

03

Comparison of rows of the matrix

In the equation \(\left[ {\begin{array}{*{20}{c}}?&?&?\\?&?&?\\?&?&?\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\\{{x_3}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{3{x_1} - 2{x_3}}\\{4{x_1}}\\{{x_1} - {x_2} + {x_3}}\end{array}} \right]\), comparing both sides of the equation, the third row of the matrix with unknown elements is \(\left[ {\begin{array}{*{20}{c}}1&{ - 1}&1\end{array}} \right]\).

So, the unknown matrix given in the equation is \(\left[ {\begin{array}{*{20}{c}}3&0&{ - 2}\\4&0&0\\1&{ - 1}&1\end{array}} \right]\).

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Most popular questions from this chapter

In Exercises 15 and 16, list five vectors in Span \(\left\{ {{v_1},{v_2}} \right\}\). For each vector, show the weights on \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\) used to generate the vector and list the three entries of the vector. Do not make a sketch.

15. \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}7\\1\\{ - 6}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 5}\\3\\0\end{array}} \right]\)

Use the accompanying figure to write each vector listed in Exercises 7 and 8 as a linear combination of u and v. Is every vector in \({\mathbb{R}^2}\) a linear combination of u and v?

8.Vectors w, x, y, and z

If Ais an \(n \times n\) matrix and the transformation \({\bf{x}}| \to A{\bf{x}}\) is one-to-one, what else can you say about this transformation? Justify your answer.

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.

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

19. There exits a matrix A such thatA[-12]=[357].
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