2.4.1: Exercises 2.4

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In Exercises $\PageIndex{1}$ – $\PageIndex{6}$, a matrix $A$ and vectors $\vec{b}$, $\vec{u}$ and $\vec{v}$ are given. Verify that $\vec{u}$ and $\vec{v}$ are both solutions to the equation $A\vec{x}=\vec{b}$; that is, show that $A\vec{u}=A\vec{v}=\vec{b}$.

Exercise $\PageIndex{1}$

$A=\left[\begin{array}{cc}{1}&{-2}\\{-3}&{6}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{0}\\{0}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{2}\\{1}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{-10}\\{-5}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

Exercise $\PageIndex{2}$

$A=\left[\begin{array}{cc}{1}&{-2}\\{-3}&{6}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{2}\\{-6}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{0}\\{-1}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{2}\\{0}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

Exercise $\PageIndex{3}$

$A=\left[\begin{array}{cc}{1}&{0}\\{2}&{0}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{0}\\{0}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{0}\\{-1}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{0}\\{59}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

Exercise $\PageIndex{4}$

$A=\left[\begin{array}{cc}{1}&{0}\\{2}&{0}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{-3}\\{-6}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{-3}\\{-1}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{-3}\\{59}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

Exercise $\PageIndex{5}$

$A=\left[\begin{array}{cccc}{0}&{-3}&{-1}&{-3}\\{-4}&{2}&{-3}&{5}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{0}\\{0}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{11}\\{4}\\{-12}\\{0}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{9}\\{-12}\\{0}\\{12}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

Exercise $\PageIndex{6}$

$A=\left[\begin{array}{cccc}{0}&{-3}&{-1}&{-3}\\{-4}&{2}&{-3}&{5}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{48}\\{36}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{-17}\\{-16}\\{0}\\{0}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{-8}\\{-28}\\{0}\\{12}\end{array}\right]$

Answer: Multiply $A\vec{u}$ and $A\vec{v}$ to verify.

In Exercises $\PageIndex{7}$ - $\PageIndex{9}$, a matrix $A$ and vectors $\vec{b}$, $\vec{u}$ and $\vec{v}$ are given. Verify that $A\vec{u}=\vec{0}$, $A\vec{v}=\vec{b}$ and $A(\vec{u}+\vec{v})=\vec{b}$.

Exercise $\PageIndex{7}$

$A=\left[\begin{array}{ccc}{2}&{-2}&{-1}\\{-1}&{1}&{-1}\\{-2}&{2}&{-1}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{1}\\{1}\\{0}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{1}\\{1}\\{-1}\end{array}\right]$

Answer: Multiply $A\vec{u}$, $A\vec{v}$ and $A(\vec{u}+\vec{v})$ to verify.

Exercise $\PageIndex{8}$

$A=\left[\begin{array}{ccc}{1}&{-1}&{3}\\{3}&{-3}&{-3}\\{-1}&{1}&{1}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{-1}\\{-3}\\{1}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{2}\\{2}\\{0}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{2}\\{3}\\{0}\end{array}\right]$

Answer: Multiply $A\vec{u}$, $A\vec{v}$ and $A(\vec{u}+\vec{v})$ to verify.

Exercise $\PageIndex{9}$

$A=\left[\begin{array}{ccc}{2}&{0}&{0}\\{0}&{1}&{-3}\\{3}&{1}&{-3}\end{array}\right],$

$\vec{b}=\left[\begin{array}{c}{2}\\{-4}\\{-1}\end{array}\right],\:\vec{u}=\left[\begin{array}{c}{0}\\{6}\\{2}\end{array}\right],\:\vec{v}=\left[\begin{array}{c}{1}\\{-1}\\{1}\end{array}\right]$

Answer: Multiply $A\vec{u}$, $A\vec{v}$ and $A(\vec{u}+\vec{v})$ to verify.

In Exercises $\PageIndex{10}$ - $\PageIndex{24}$, a matrix $A$ and vector $\vec{b}$ are given.

Solve the equation $A\vec{x}=\vec{0}$.
Solve the equation $A\vec{x}=\vec{b}$.

In each of the above, be sure to write your answer in vector format. Also, when possible, give 2 particular solutions to each equation.

Exercise $\PageIndex{10}$

$A=\left[\begin{array}{cc}{0}&{2}\\{-1}&{3}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-2}\\{-1}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-2}\\{-1}\end{array}\right]$

Exercise $\PageIndex{11}$

$A=\left[\begin{array}{cc}{-4}&{-1}\\{-3}&{-2}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{1}\\{4}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{2/5}\\{-13/5}\end{array}\right]$

Exercise $\PageIndex{12}$

$A=\left[\begin{array}{cc}{1}&{-2}\\{0}&{1}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{0}\\{-5}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-10}\\{-5}\end{array}\right]$

Exercise $\PageIndex{13}$

$A=\left[\begin{array}{cc}{1}&{0}\\{5}&{-4}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-2}\\{-1}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-2}\\{-9/4}\end{array}\right]$

Exercise $\PageIndex{14}$

$A=\left[\begin{array}{cc}{2}&{-3}\\{-4}&{6}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{1}\\{-1}\end{array}\right]$

Answer

$\vec{x}=x_{2}\left[\begin{array}{c}{3/2}\\{1}\end{array}\right]$
No solution.

Exercise $\PageIndex{15}$

$A=\left[\begin{array}{ccc}{-4}&{3}&{2}\\{-4}&{5}&{0}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-4}\\{-4}\end{array}\right]$

Answer

$\vec{x}=x_{3}\left[\begin{array}{c}{5/4}\\{1}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]+x_{3}\left[\begin{array}{c}{5/4}\\{1}\\{1}\end{array}\right]$

Exercise $\PageIndex{16}$

$A=\left[\begin{array}{ccc}{1}&{5}&{-2}\\{1}&{4}&{5}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{0}\\{1}\end{array}\right]$

Answer

$\vec{x}=x_{3}\left[\begin{array}{c}{-33}\\{7}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{5}\\{-1}\\{0}\end{array}\right]+x_{3}\left[\begin{array}{c}{-33}\\{7}\\{1}\end{array}\right]$

Exercise $\PageIndex{17}$

$A=\left[\begin{array}{ccc}{-1}&{-2}&{-2}\\{3}&{4}&{-2}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-4}\\{-4}\end{array}\right]$

Answer

$\vec{x}=x_{3}\left[\begin{array}{c}{14}\\{-10}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-4}\\{2}\end{array}\right]+x_{3}\left[\begin{array}{c}{14}\\{-10}\\{0}\end{array}\right]$

Exercise $\PageIndex{18}$

$A=\left[\begin{array}{ccc}{2}&{2}&{2}\\{5}&{5}&{-3}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{3}\\{-3}\end{array}\right]$

Answer

$\vec{x}=x_{2}\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{3/16}\\{0}\\{21/16}\end{array}\right]+x_{2}\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]$

Exercise $\PageIndex{19}$

$A=\left[\begin{array}{cccc}{1}&{5}&{-4}&{-1}\\{1}&{0}&{-2}&{1}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{0}\\{-2}\end{array}\right]$

Answer

$\vec{x}=x_{3}\left[\begin{array}{c}{2}\\{2/5}\\{1}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{-1}\\{2/5}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-2}\\{2/5}\\{0}\\{0}\end{array}\right]+x_{3}\left[\begin{array}{c}{2}\\{2/5}\\{1}\\{0}\end{array}\right]+\left[\begin{array}{c}{-1}\\{2/5}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{20}$

$A=\left[\begin{array}{cccc}{-4}&{2}&{-5}&{4}\\{0}&{1}&{-1}&{5}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-3}\\{-2}\end{array}\right]$

Answer

$\vec{x}=x_{3}\left[\begin{array}{c}{-3/4}\\{1}\\{1}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{-3/2}\\{-5}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-1/4}\\{-2}\\{0}\\{0}\end{array}\right]+x_{3}\left[\begin{array}{c}{-3/4}\\{1}\\{1}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{-3/2}\\{-5}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{21}$

$A=\left[\begin{array}{ccccc}{0}&{0}&{2}&{1}&{4}\\{-2}&{-1}&{-4}&{-1}&{5}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{3}\\{4}\end{array}\right]$

Answer

$\vec{x}=x_{2}\left[\begin{array}{c}{-1/2}\\{1}\\{0}\\{0}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{1/2}\\{0}\\{-1/2}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{13/2}\\{0}\\{-2}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{-5}\\{0}\\{3/2}\\{0}\\{0}\end{array}\right]+x_{2}\left[\begin{array}{c}{-1/2}\\{1}\\{0}\\{0}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{1/2}\\{0}\\{-1/2}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{13/2}\\{0}\\{-2}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{22}$

$A=\left[\begin{array}{ccccc}{3}&{0}&{-2}&{-4}&{5}\\{2}&{3}&{2}&{0}&{2}\\{-5}&{0}&{4}&{0}&{5}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-1}\\{-5}\\{4}\end{array}\right]$

Answer

$\vec{x}=x_{4}\left[\begin{array}{c}{8}\\{-12}\\{10}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{-15}\\{68/3}\\{-20}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{2}\\{-16/3}\\{7/2}\\{0}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{8}\\{-12}\\{10}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{-15}\\{68/3}\\{-20}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{23}$

$A=\left[\begin{array}{ccccc}{-1}&{3}&{1}&{-3}&{4}\\{3}&{-3}&{-1}&{1}&{-4}\\{-2}&{3}&{-2}&{-3}&{1}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{1}\\{1}\\{-5}\end{array}\right]$

Answer

$\vec{x}=x_{4}\left[\begin{array}{c}{1}\\{13/9}\\{-1/3}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{0}\\{-1}\\{-1}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{1}\\{1/9}\\{5/3}\\{0}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{1}\\{13/9}\\{-1/3}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{0}\\{-1}\\{-1}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{24}$

$A=\left[\begin{array}{ccccc}{-4}&{-2}&{-1}&{4}&{0}\\{5}&{-4}&{3}&{-1}&{1}\\{4}&{-5}&{3}&{1}&{-4}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{3}\\{2}\\{1}\end{array}\right]$

Answer

$\vec{x}=x_{4}\left[\begin{array}{c}{3}\\{-1}\\{-6}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{-17}\\{12}\\{44}\\{0}\\{1}\end{array}\right]$
$\vec{x}=\left[\begin{array}{c}{7}\\{-6}\\{-19}\\{0}\\{0}\end{array}\right]+x_{4}\left[\begin{array}{c}{3}\\{-1}\\{-6}\\{1}\\{0}\end{array}\right]+x_{5}\left[\begin{array}{c}{-17}\\{12}\\{44}\\{0}\\{1}\end{array}\right]$

In Exercises $\PageIndex{25}$ – $\PageIndex{28}$, a matrix $A$ and vector $\vec{b}$ are given. Solve the equation $A\vec{x}=\vec{b}$, write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane.

Exercise $\PageIndex{25}$

$A=\left[\begin{array}{cc}{2}&{4}\\{-1}&{-2}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$

Answer

$\vec{x}=x_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]=x_{2}\vec{v}$

$clipboard_e46b1e81e74ff69bc1efa2b1870096469.png$

Figure $\PageIndex{1}$

Exercise $\PageIndex{26}$

$A=\left[\begin{array}{cc}{2}&{4}\\{-1}&{-2}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{-6}\\{3}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{-3}\\{0}\end{array}\right]+x_{2}\left[\begin{array}{c}{-2}\\{1}\end{array}\right]=\vec{x_{p}}+x_{2}\vec{v}$

$clipboard_e7d4233acc2411f64fc8a317bbb81083d.png$

Figure $\PageIndex{2}$

Exercise $\PageIndex{27}$

$A=\left[\begin{array}{cc}{2}&{-5}\\{-4}&{-10}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{1}\\{2}\end{array}\right]$

Answer

$\vec{x}=\left[\begin{array}{c}{0.5}\\{0}\end{array}\right]+x_{2}\left[\begin{array}{c}{2.5}\\{1}\end{array}\right]=\vec{x_{p}}+x_{2}\vec{v}$

$clipboard_ea192547b7ad69a2b0fd1db5a3a424ecd.png$

Figure $\PageIndex{3}$

Exercise $\PageIndex{28}$

$A=\left[\begin{array}{cc}{2}&{-5}\\{-4}&{-10}\end{array}\right],\:\vec{b}=\left[\begin{array}{c}{0}\\{0}\end{array}\right]$

Answer

$\vec{x}=x_{2}\left[\begin{array}{c}{2.5}\\{1}\end{array}\right]=x_{2}\vec{v}$

$clipboard_e2f219edc40fad5b66c9b65f8d39bc499.png$

Figure $\PageIndex{4}$

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