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Chapter 1: Q25E (page 1)
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
The given coefficient matrix is consistent.
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Consider the problem of determining whether the following system of equations is consistent:
\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)
Follow the method of Example 3 to describe the solutions of the following system in parametric vector form. Also, give a geometric description of the solution set and compare it to that in Exercise 5.
\(\begin{aligned}{c}{x_1} + 3{x_2} + {x_3} = 1\\ - 4{x_1} - 9{x_2} + 2{x_3} = - 1\\ - 3{x_2} - 6{x_3} = - 3\end{aligned}\)
In a grid of wires, the temperature at exterior mesh points is maintained at constant values, as shown in the accompanying figure. When the grid is in thermal equilibrium, the temperature Tat each interior mesh point is the average of the temperatures at the four adjacent points. For example,
Find the temperatures andwhen the grid is in thermal equilibrium.
Determine the values(s) of \(h\) such that matrix is the augmented matrix of a consistent linear system.
18. \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&{ - 2}\\5&h&{ - 7}\end{array}} \right]\)
In Exercise 23 and 24, mark each statement True or False. Justify each answer.
23.
a. A homogeneous equation is always consistent.
b. The equation \(Ax = 0\) gives an explicit description of its solution set.
c. The homogeneous equation \(Ax = 0\) has the trivial solution if and only if the equation has at least one free variable.
d. The equation \(x = p + tv\) describes a line through \({\mathop{\rm v}\nolimits} \) parallel to \(p\).
e. The solution set of \(Ax = b\) is the set of all vectors of the form \({\mathop{\rm w}\nolimits} = p + {v_k}\), where \({v_k}\) is any solution of the equation \(Ax = 0\).
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