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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

Write the reduced echelon form of a \(3 \times 3\) matrix A such that the first two columns of Aare pivot columns and

\(A = \left( {\begin{aligned}{*{20}{c}}3\\{ - 2}\\1\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}0\\0\\0\end{aligned}} \right)\).

Describe the possible echelon forms of the matrix A. Use the notation of Example 1 in Section 1.2.

a. A is a \({\bf{2}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{2}}}\).

b. A is a \({\bf{3}} \times {\bf{3}}\) matrix whose columns span \({\mathbb{R}^{\bf{3}}}\).

Find the general solutions of the systems whose augmented matrices are given as

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

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. (Hint: Given u, v in \({\mathbb{R}^n}\), let \({\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\). Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \(T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \). Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).)

Solve the systems in Exercises 11‑14.

12.\(\begin{aligned}{c}{x_1} - 3{x_2} + 4{x_3} = - 4\\3{x_1} - 7{x_2} + 7{x_3} = - 8\\ - 4{x_1} + 6{x_2} - {x_3} = 7\end{aligned}\)
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