Linear Algebra and Its Applications, 5th Edition
Authors: David C. Lay, Steven R. Lay, Judi J. McDonald
ISBN-13: 978-0321982384
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See our solution for Question 2E from Chapter 1.10 from Lay's Linear Algebra and Its Applications, 5th Edition.
Problem 2E
Chapter:
Problem:
0
Step-by-Step Solution
Given Information
We are given with following Data:
Step-1: The Initial Matrix
The matrix B and vector U is: \[B = \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \text { and } u = \left[ \begin{array} { l } { 3 } \\ { 2 } \end{array} \right]\]The amounts of calories, protein, fiber, and fat contained in a mixture of three servings of Shredded Wheat and two servings of Crispix are: \[\begin{aligned} B u & = \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \left[ \begin{array} { l } { 3 } \\ { 2 } \end{array} \right] \\ & = 3 \left[ \begin{array} { c } { 160 } \\ { 5 } \\ { 6 } \\ { 1 } \end{array} \right] + 2 \left[ \begin{array} { l } { 110 } \\ { 2 } \\ { 0.1 } \\ { 0.4 } \end{array} \right] \end{aligned}\]
Step-2:
Let, \[u = \left[ \begin{array} { l } { x } \\ { y } \end{array} \right]\]
Step-3:
\[\begin{array} { r } { B u = \left[ \begin{array} { l } { 130 } \\ { 3.2 } \\ { 2.46 } \\ { 0.64 } \end{array} \right] } \\ { \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \left[ \begin{array} { l } { x } \\ { y } \end{array} \right] = \left[ \begin{array} { c } { 130 } \\ { 3.2 } \\ { 2.46 } \\ { 0.64 } \end{array} \right] } \end{array}\]
Step-4: The Augmented Matrix
\[[ B , b ] = \left[ \begin{array} { c c c } { 160 } & { 110 } & { 130 } \\ { 5 } & { 2 } & { 3.2 } \\ { 6 } & { 0.1 } & { 2.46 } \\ { 1 } & { 0.4 } & { 0.64 } \end{array} \right]\]Row-Reduce the Augmented Matrix using MATLAB: \[[B,b] = \left[ {\begin{array}{*{20}{c}}{160}&{110}&{130}\\5&2&{3.2}\\6&{0.1}&{2.46}\\1&{0.4}&{0.64}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0&{0.4}\\0&1&{0.6}\\0&0&0\\0&0&0\end{array}} \right]\]Thus, the solution for u is: \[u = \left[ {\begin{array}{*{20}{c}}x\\y\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{0.4}\\{0.6}\end{array}} \right]\]Therefore,
0.4 unit of Shredded Wheat and 0.6 unit of Crispix give the required Nutrient.
We are given with following Data:
| Nutrient | Post Shredded Wheat | Crispix |
|---|---|---|
| Calories | 160 | 110 |
| Protein(g) | 5 | 2 |
| Fiber(g) | 6 | 0.1 |
| Fat(g) | 1 | 0.4 |
Step-1: The Initial Matrix
The matrix B and vector U is: \[B = \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \text { and } u = \left[ \begin{array} { l } { 3 } \\ { 2 } \end{array} \right]\]The amounts of calories, protein, fiber, and fat contained in a mixture of three servings of Shredded Wheat and two servings of Crispix are: \[\begin{aligned} B u & = \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \left[ \begin{array} { l } { 3 } \\ { 2 } \end{array} \right] \\ & = 3 \left[ \begin{array} { c } { 160 } \\ { 5 } \\ { 6 } \\ { 1 } \end{array} \right] + 2 \left[ \begin{array} { l } { 110 } \\ { 2 } \\ { 0.1 } \\ { 0.4 } \end{array} \right] \end{aligned}\]
Step-2:
Let, \[u = \left[ \begin{array} { l } { x } \\ { y } \end{array} \right]\]
Step-3:
\[\begin{array} { r } { B u = \left[ \begin{array} { l } { 130 } \\ { 3.2 } \\ { 2.46 } \\ { 0.64 } \end{array} \right] } \\ { \left[ \begin{array} { c c } { 160 } & { 110 } \\ { 5 } & { 2 } \\ { 6 } & { 0.1 } \\ { 1 } & { 0.4 } \end{array} \right] \left[ \begin{array} { l } { x } \\ { y } \end{array} \right] = \left[ \begin{array} { c } { 130 } \\ { 3.2 } \\ { 2.46 } \\ { 0.64 } \end{array} \right] } \end{array}\]
Step-4: The Augmented Matrix
\[[ B , b ] = \left[ \begin{array} { c c c } { 160 } & { 110 } & { 130 } \\ { 5 } & { 2 } & { 3.2 } \\ { 6 } & { 0.1 } & { 2.46 } \\ { 1 } & { 0.4 } & { 0.64 } \end{array} \right]\]Row-Reduce the Augmented Matrix using MATLAB: \[[B,b] = \left[ {\begin{array}{*{20}{c}}{160}&{110}&{130}\\5&2&{3.2}\\6&{0.1}&{2.46}\\1&{0.4}&{0.64}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0&{0.4}\\0&1&{0.6}\\0&0&0\\0&0&0\end{array}} \right]\]Thus, the solution for u is: \[u = \left[ {\begin{array}{*{20}{c}}x\\y\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{0.4}\\{0.6}\end{array}} \right]\]Therefore,
0.4 unit of Shredded Wheat and 0.6 unit of Crispix give the required Nutrient.