Chapter 9

In Problems 29-34, determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} x-2 y+z=3 \\ 2 x-3 y+z=8 \end{array} $$

In Problems 29-34, let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } \mathbf{u}+\mathbf{v} \text { and illustrate the result graphically. } $$

Let \(\mathbf{x}=[2,0,-1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

In Problems , let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } \mathbf{u}-\mathbf{v} \text { and illustrate the result graphically. } $$

Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ 0 & -2 \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{lllr} 1 & 2 & 0 & -1 \\ 2 & 1 & 3 & 0 \end{array}\right] $$ (a) Compute \(A B\). (b) Compute \(B^{\prime} A\).

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 2 x-y=3 \\ x-y=4 \\ 3 x-y=1 \end{array} $$

A triangle has vertices at coordinates \(P=(0,0), Q=(4,0)\), and \(R=(4,3)\). (a) Use basic trigonometry to compute the lengths of all three sides and the measures of all three angles. (b) Use the results of this section to repeat (a).

In Problems , let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } \mathbf{w}-\mathbf{u} \text { and illustrate the result graphically. } $$

Let \(A=\left[\begin{array}{lll}1 & 4 & -2\end{array}\right]\) and \(B=\left[\begin{array}{r}-1 \\ 2 \\ 3\end{array}\right]\) (a) Compute \(A B\). (b) Compute \(B A\).

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 4 y-3 z=6 \\ 2 y+z=1 \\ y+z=0 \end{array} $$

In Problems , let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } \mathbf{v}-\frac{1}{2} \mathbf{u} \text { and illustrate the result graphically. } $$

33\. Let $$ A=\left[\begin{array}{rr} 2 & 1 \\ -1 & -3 \end{array}\right] $$ Find \(A^{2}, A^{3}\), and \(A^{4}\).

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 2 x-7 y+z=2 \\ x+y-2 z=4 \end{array} $$

A triangle has vertices at coordinates \(P=(1,2,3), Q=\) \((1,5,2)\), and \(R=(2,4,2)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

Suppose that $$ A=\left[\begin{array}{rr} 1 & -1 \\ 3 & 0 \\ 5 & 2 \end{array}\right] \text { and } B=\left[\begin{array}{lll} 2 & 4 & 1 \\ 6 & 0 & 0 \end{array}\right] $$ Show that \((A B)^{\prime}=B^{\prime} A^{\prime}\).

Determine whether each system is overdetermined or underdetermined; then solve each system. $$ \begin{array}{r} 3 x+y=1 \\ x-y=0 \\ 4 x \quad=1 \end{array} $$

A triangle has vertices at coordinates \(P=(2,1,5), Q=\) \((-1,-3,7)\), and \(R=(2,-4,1)\) (a) Compute the lengths of all three sides. (b) Compute all three angles in both radians and degrees.

In Problems , let $$ \mathbf{u}=\left[\begin{array}{l} 3 \\ 4 \end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l} -1 \\ -2 \end{array}\right], \quad \text { and } \quad \mathbf{w}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] $$ $$ \text { Compute } 2 \mathbf{v}-\mathbf{w} \text { and illustrate the result graphically. } $$

Let $$ B=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right] $$ (a) Find \(B^{2}, B^{3}, B^{4}\), and \(B^{5}\). (b) What can you say about \(B^{k}\) when (i) \(k\) is even and (ii) \(k\) is odd?

SplendidLawn sells three types of lawn fertilizer: SL \(24-4-\) 8, SL 21-7-12 and SL \(17-0-0 .\) The three numbers refer to the percentages of nitrogen, phosphate, and potassium, in that order, of the contents. (For instance, \(100 \mathrm{~g}\) of SL 24-4-8 contains \(24 \mathrm{~g}\) of nitrogen.) Suppose that each year your lawn requires \(500 \mathrm{~g}\) of nitrogen, \(100 \mathrm{~g}\) of phosphate, and \(180 \mathrm{~g}\) of potassium per 1000 square feet. How much of each of the three types of fertilizer should you apply per 1000 square feet per year?

$$ \begin{array}{l} \text { In Problems 35-40, give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$

Let $$ I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ Show that \(I_{3}=I_{3}^{2}=I_{3}^{3}\).

Three different species of insects are reared together in a laboratory cage. They are supplied with two different types of food each day. Each individual of species 1 consumes 3 units of food \(A\) and 5 units of food \(B\), each individual of species 2 consumes 2 units of food \(A\) and 3 units of food \(B\), and individual of species 3 consumes 1 unit of food \(A\) and 2 units of food \(B\). Each day, 500 units of food \(A\) and 900 units of food \(B\) are supplied. How many individuals of each species can be reared together? Is there more than one solution? What happens if we add 550 units of a third type of food, called \(C\), and each individual of species 1 consumes 2 units of food \(C\), each individual of species 2 consumes 4 units of food \(C\), and each individual of species 3 consumes 1 unit of food \(C ?\)

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\left[\begin{array}{rr} 2 & 0 \\ 0 & -1 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ 0 & -2 \end{array}\right] \text { and } I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $$ Show that \(A I_{2}=I_{2} A=A\).

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\left[\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rrr} 1 & 3 & 0 \\ 0 & -1 & 2 \\ -1 & -2 & 1 \end{array}\right] \text { and } I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ Show that \(A I_{3}=I_{3} A=A\).

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\left[\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right] $$

Write each system in matrix form. $$ \begin{array}{r} 2 x_{1}+3 x_{2}-x_{3}=0 \\ 2 x_{2}+x_{3}=1 \\ x_{1} \quad-2 x_{3}=2 \end{array} $$

Find the equation of the plane through \((1,2,3)\) and perpendicular to \([0,-1,1]^{\prime}\).

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\frac{1}{2}\left[\begin{array}{rr} \sqrt{3} & -1 \\ 1 & \sqrt{3} \end{array}\right] $$

Write each system in matrix form. $$ \begin{array}{r} 2 x_{2}-x_{1}=x_{3} \\ 4 x_{1}+x_{3}=7 x_{2} \\ x_{2}-x_{1}=x_{3} \end{array} $$

Find the equation of the plane through \((1,0,-3)\) and perpendicular to \([1,-2,-1]^{\prime}\).

$$ \begin{array}{l} \text { In Problems , give a geometric interpretation of the map }\\\ \mathbf{x} \rightarrow A \mathbf{x} \text { for each given map } A . \end{array} $$ $$ A=\frac{1}{2}\left[\begin{array}{lr} \sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2} \end{array}\right] $$

$$ \begin{array}{r} 2 x_{1}-3 x_{2}=4 \\ -x_{1}+x_{2}=3 \\ 3 x_{1}=4 \end{array} $$

Find the equation of the plane through \((0,0,0)\) and perpendicular to \([1,0,0]^{\prime}\).

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}-1 \\\ 2\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Write each system in matrix form. $$ \begin{array}{r} x_{1}-2 x_{2}+x_{3}=1 \\ -2 x_{1}+x_{2}-3 x_{3}=0 \end{array} $$

Find the equation of the plane through \((3,-1,2)\) and perpendicular to \([-1,1,2]^{\prime} .\)

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}4 \\\ -1\end{array}\right]\) counterclockwise by the angle \(\pi / 3\).

Show that the inverse of $$ A=\left[\begin{array}{ll} 2 & 1 \\ 1 & 1 \end{array}\right] $$ is $$ B=\left[\begin{array}{rr} 1 & -1 \\ -1 & 2 \end{array}\right] $$

In Problems \(43-46\), find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$ (1,-1),\left[\begin{array}{l} 2 \\ 1 \end{array}\right] $$

Use a rotation matrix to rotate the vector \begin{tabular}{|l|l} 5 & counterclock- \\ 2 \end{tabular} wise by the angle \(\pi / 12\).

Show that the inverse of $$ A=\left[\begin{array}{lll} 2 & 3 & 1 \\ 5 & 2 & 3 \\ 1 & 2 & 0 \end{array}\right] $$ is $$ B=\left[\begin{array}{rrr} -\frac{6}{5} & \frac{2}{5} & \frac{7}{5} \\ \frac{3}{5} & -\frac{1}{5} & -\frac{1}{5} \\ \frac{8}{5} & -\frac{1}{5} & -\frac{11}{5} \end{array}\right] $$

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$ (3,-4),\left[\begin{array}{r} -1 \\ 2 \end{array}\right] $$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}-2 \\\ -3\end{array}\right]\) counterclockwise by the angle \(\pi / 9\).

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of \(A\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$ (-1,-2),\left[\begin{array}{r} 1 \\ -3 \end{array}\right] $$

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of \(B\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$ (-1,4),\left[\begin{array}{l} 2 \\ 3 \end{array}\right] $$