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Chapter 1: Q18E (page 39)
Determine whether the statements that follow are true or false, and justify your answer.
18:
False, because the product of , which is not same as the given product.
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Find the general solutions of the systems whose augmented matrices are given in Exercises 10.
10. \(\left[ {\begin{array}{*{20}{c}}1&{ - 2}&{ - 1}&3\\3&{ - 6}&{ - 2}&2\end{array}} \right]\)
An important concern in the study of heat transfer is to determine the steady-state temperature distribution of a thin plate when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross section of a metal beam, with negligible heat flow in the direction perpendicular to the plate. Let \({T_1},...,{T_4}\) denote the temperatures at the four interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of the four nearest nodes鈥攖o the left, above, to the right, and below. For instance,
\({T_1} = \left( {10 + 20 + {T_2} + {T_4}} \right)/4\), or \(4{T_1} - {T_2} - {T_4} = 30\)
33. Write a system of four equations whose solution gives estimates
for the temperatures \({T_1},...,{T_4}\).
In Exercises 6, write a system of equations that is equivalent to the given vector equation.
6. \({x_1}\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\end{array}} \right] + {x_2}\left[ {\begin{array}{*{20}{c}}8\\5\end{array}} \right] + {x_3}\left[ {\begin{array}{*{20}{c}}1\\{ - 6}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right]\)
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 \({\mathbb{R}^{\bf{3}}}\).
Determine h and k such that the solution set of the system (i) is empty, (ii) contains a unique solution, and (iii) contains infinitely many solutions.
a. \({x_1} + 3{x_2} = k\)
\(4{x_1} + h{x_2} = 8\)
b. \( - 2{x_1} + h{x_2} = 1\)
\(6{x_1} + k{x_2} = - 2\)
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