1.
\(\left(\begin{array}{rrr} 3\amp -6\amp 2\\8\amp 5\amp -4\\6\amp 0\amp -1 \end{array} \right)\begin{pmatrix}x\\y\\z \end{pmatrix} =\left(\begin{array}{r}-16\\32\\11 \end{array} \right)\text{.}\)
Section 3.3 Elimination: Homework
Exercises Exercises
Exercise Group.
If possible, solve the given matrix-vector equation \(A\vec{x}=\vec{b}\text{.}\) If not, explain. Be sure to label your row operations over (and under if necessary) your arrows.
2.
\(\left(\begin{array}{rrr} 3\amp -6\amp 2\\8\amp 5\amp -4\\6\amp 0\amp -1 \end{array} \right)\begin{pmatrix}x\\y\\z \end{pmatrix} =\left(\begin{array}{r}-56\\21\\-17 \end{array} \right)\text{.}\)
Exercise Group.
If possible, solve the system represented by the given augmented matrix. If not, explain. Be sure to label your row operations over (and under if necessary) your arrows.
3.
\(\left(\begin{array}{rrr|r}-2\amp 4\amp 1\amp 4\\3\amp 0\amp -7\amp -33\\-2\amp 16\amp -10\amp -49 \end{array} \right)\)
4.
\(\left(\begin{array}{rrr|r}-2\amp 4\amp 1\amp 2\\3\amp 0\amp -7\amp 5\\-2\amp 16\amp -10\amp 13 \end{array} \right)\)
5.
\(\left(\begin{array}{rrr|r}-2\amp 4\amp 1\amp 1\\3\amp 0\amp -7\amp 4\\-2\amp 16\amp -10\amp 12 \end{array} \right)\)
6.
\(\left(\begin{array}{rrr|r}-2\amp 4\amp 1\amp 2\\3\amp 0\amp -7\amp 5\\-2\amp 16\amp -10\amp 18 \end{array} \right)\)
7.
A system of linear equations cannot have exactly two solutions! Show why this is true by assuming that \(\begin{pmatrix}x_1\\y_1\\z_1 \end{pmatrix}\) and \(\begin{pmatrix}x_2\\y_2\\z_2 \end{pmatrix}\) both solve \(A\vec{x}=\vec{b}\text{,}\) and find a third solution \(\begin{pmatrix}x_3\\y_3\\z_3 \end{pmatrix} \text{.}\) (This means that you can find lots more, by mathematical induction.)
Exercise Group.
For which three values of \(c\) will elimination on the given matrix fail to produce a full set of pivots? For each \(c\) that you found, explain why elimination on the resulting matrix fails to produce a full set of pivots.
8.
\(\begin{pmatrix}c\amp 1\amp 2\\c\amp c\amp 4\\c\amp c\amp c \end{pmatrix}\)
9.
\(\begin{pmatrix}c\amp 5\amp 3\\c\amp c\amp 7\\c\amp c\amp c \end{pmatrix}\)
10.
Choose any two parallel distinct lines in \(\R^2\) and write down their equations. Then rewrite the equations as a linear system \(\A\x=\b\) and attempt to solve by elimination. What happens?11.
Choose any two lines in \(\R^2\) with different slopes and write down their equations. Then rewrite the equations as a linear system \(\A\x=\b\) and attempt to solve by elimination. What happens?12.
Suppose \(X,Y\in\) the set of all people, that \(X\) is four times as old as \(Y\text{,}\) and that their ages add to 120. What are their ages? Write the problem as a linear system and solve it.
13.
The points \((2,5)\) and \((3,8)\) lie on the same line. Which line? Using the slope-intercept form of the equation for a line in \(\R^2\text{,}\) write the problem as a linear system and solve it.
14.
Create a matrix \(B\) with all nonzero entries, and in which no row is a multiple of any other row, but for which elimination produces a zero row.
15.
Create a matrix \(B\) with all nonzero entries, and in which no row is a multiple of any other row, but for which elimination does not produce a zero row.
16.
Consider the system \(\left(\begin{array}{rrr|r}2 \amp 3 \amp 4 \amp 5\\-3 \amp 0 \amp 8 \amp -6\\a \amp b \amp c \amp d\end{array}\right).\)
- Find \(a,b,c,d\in\R\setminus\{0\}\text{,}\) with \((a,b,c)\) is not a multiple of the first nor the second row, so that the system is singular and consistent.
- Find \(a,b,c,d\in\R\setminus\{0\}\text{,}\) with \((a,b,c)\) is not a multiple of the first nor the second row, so that the system is singular and inconsistent.
17.
Suppose \(A\in\R^{3\times 3}\) has \(3R_1-4R_2=R_3\text{.}\)
- Which right sides \(\vec{b}=\begin{pmatrix}b_1\\b_2\\b_3 \end{pmatrix}\) produce a system \(A\vec{x}=\vec{b}\) with a solution?
- Why can’t \(A\vec{x}=\begin{pmatrix}1\\2\\3\end{pmatrix}\) for any \(\vec{x}\in\R^3\text{?}\)
18.
Prove Theorem 3.1.38.
