Systems of Equations and Inequalities

In Problems 41– 60, use the inverses found in Problems 31–40 to solve each system of equations

$\left\{\begin{array}{l}x+y+z=2\\ 3x+2y-z=\frac{7}{3}\\ 3x+y+2z=\frac{10}{3}\end{array}\right.$

In Problems 41– 60, use the inverses found in Problems 31–40 to solve each system of equation 

$\left\{\begin{array}{r}3x+3y+z=1\\ x+2y+z=0\\ 2x-y+z=4\end{array}\right.$

In Problems 61– 66, show that each matrix has no inverse. 

$\left[\begin{array}{ll}4& 2\\ 2& 1\end{array}\right]$

In Problems 61– 66, show that each matrix has no inverse. 

$\left[\begin{array}{rr}-3& \frac{1}{2}\\ 6& -1\end{array}\right]$

In Problems 61– 66, show that each matrix has no inverse. 

$\left[\begin{array}{ll}15& 3\\ 10& 2\end{array}\right]$

In Problems 61– 66, show that each matrix has no inverse 

$\left[\begin{array}{rr}-3& 0\\ 4& 0\end{array}\right]$

In Problems 61– 66, show that each matrix has no inverse 

$\left[\begin{array}{rrr}-3& 1& -1\\ 1& -4& -7\\ 1& 2& 5\end{array}\right]$

 Show that each matrix has no inverse. 

 Use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.

$\left[\begin{array}{ccc}25& 61& -12\\ 18& -2& 4\\ 8& 35& 21\end{array}\right]$.

use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.

$\left[\begin{array}{ccc}18& -3& 4\\ 6& -20& 14\\ 10& 25& -15\end{array}\right]$

use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.

$\left[\begin{array}{cccc}44& 21& 18& 6\\ -2& 10& 15& 5\\ 21& 12& -12& 4\\ -8& -16& 4& 9\end{array}\right]$

use a graphing utility to find the inverse, if it exists, of each matrix. Round answers to two decimal places.

$\left[\begin{array}{cccc}16& 22& -3& 5\\ 21& -17& 4& 8\\ 2& 8& 27& 30\\ 5& 15& -3& -10\end{array}\right]$

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=10\\ 18x-12y+7z=-9\\ 3x+4y-z=12\end{array}\right.$

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=15\\ 18x-12y+7z=-3\\ 3x+4y-z=12\end{array}\right.$

Use the idea behind Example 15 with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=21\\ 18x-12y+7z=7\\ 3x+4y-z=-2\end{array}\right.$

Use the idea behind Example  with a graphing utility to solve the following systems of equations. Round answers to two decimal places.

$\left\{\begin{array}{l}25x+61y-12z=25\\ 18x-12y+7z=10\\ 3x+4y-z=-4\end{array}\right.$

Algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}2x+3y=11\\ 5x+7y=24\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}2x+8y=-8\\ x+7y=-13\end{array}\right.$

In Problems 75– 8 2, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}x-2y+4z=2\\ -3x+5y-2z=17\\ 4x-3y=-22\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}2x+3y-z=-2\\ 4x+3z=6\\ 6y-2z=2\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}5x-y+4z=2\\ -x+5y-4z=3\\ 7x+13y-4z=17\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}3x+2y-z=2\\ 2x+y+6z=-7\\ 2x+2y-14z=17\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}2x-3y+z=4\\ -3x+2y-z=17\\ -5y+z=6\end{array}\right.$

In Problems 75– 82, algebraically solve each system of equations using any method you wish.

$\left\{\begin{array}{l}-4x+3y+2x=6\\ 3x+y-z=-2\\ x+9y+z=6\end{array}\right.$

College Tuition Nikki and Joe take classes at a community college, LCCC, and a local university, SIUE. The number of credit hours taken and the cost per credit hour (2009–2010

academic year, tuition only) are as follows:

(a) Write a matrix A for the credit hours taken by each student and a matrix B for the cost per credit hour.

(b) Compute AB and interpret the results.

School Loan Interest Jamal and Stephanie each have school loans issued from the same two banks. The amounts borrowed and the monthly interest rates are given next (interest is compounded monthly):

(a) Write a matrix A for the amounts borrowed by each student and a matrix B for the monthly interest rates.

(b) Compute AB and interpret the results.

(c) Let $C=\left[\begin{array}{c}1\\ 1\end{array}\right]$. Compute $A(C+B)$and interpret the results.

Computing the Cost of Production The Acme Steel    Company is a producer of stainless steel and aluminum containers. On a certain day, the following stainless steel containers were manufactured: 500 with 10-gallon (gal) capacity, 350 with 5-gal capacity, and 400 with 1-gal capacity. On the same day, the following aluminum containers were manufactured: 700 with 10-gal capacity, 500 with 5-gal capacity, and 850 with 1-gal capacity.

(a) Find a 2 by 3 matrix representing these data. Find a 3 by 2 matrix to represent the same data. 

(b) If the amount of material used in the 10-gal containers is 15 pounds (lb), the amount used in the 5-gal containers is 8 lb, and the amount used in the 1-gal containers is 3 lb, find a 3 by 1 matrix representing the amount of material used.

(c) Multiply the 2 by 3 matrix found in part (a) and the 3 by 1 matrix found in part (b) to get a 2 by 1 matrix showing the day’s usage of material.

(d) If stainless steel costs Acme \(0.10 lb and aluminum costs \)0.05 lb, find a 1 by 2 matrix representing cost.

(e) Multiply the matrices found in parts (c) and (d) to determine the total cost of the day’s production.

Computing Profit; Rizza Ford has two locations, one in the city and the other in the suburbs. In January, the city location sold $400$ subcompacts, $250$ intermediate-size cars, and $50$ SUVs; in February, it sold $350$ subcompacts, $100$ intermediates, and $30$ SUVs. At the suburban location in January, $450$ subcompacts, $200$ intermediates, and $140$ SUVs were sold. In February, the suburban location sold $350$ subcompacts, $300$ intermediates, and $100$ SUVs.

(a) Find $2$ by $3$ matrices that summarize the sales data for each location for January and February (one matrix for each month).

(b) Use matrix addition to obtain total sales for the $2$-month period.

(c) The profit on each kind of car is $\backslash (100$ per subcompact, $\backslash )150$ per intermediate, and $\$200$ per SUV. Find a $3$ by $1$ matrix representing this profit.

(d) Multiply the matrices found in parts (b) and (c) to get a $2$ by $1$ matrix showing the profit at each location 

Cryptography; One method of encryption is to use a matrix to encrypt the message and then use the corresponding inverse matrix to decode the message. The encrypted matrix, $E$, is obtained by multiplying the message matrix, $M$, by a key matrix, $K$. The original message can be retrieved by multiplying the encrypted matrix by the inverse of the key matrix. That is, $E=M·K$ and $M=E·K^{-1}$.

(a) Given the key matrix $K=\left[\begin{array}{ccc}2& 1& 1\\ 1& 1& 0\\ 1& 1& 1\end{array}\right]$ , find its inverse, $K^{-1}$. [Note: This key matrix is known as the ${Q_2}^3$ Fibonacci encryption matrix.]

(b) Use your result from part (a) to decode the encrypted matrix $E=\left[\begin{array}{ccc}47& 34& 33\\ 44& 36& 27\\ 47& 41& 20\end{array}\right]$

(c) Each entry in your result for part (b) represents the position of a letter in the English alphabet ($A = 1, B = 2,C = 3,$ and so on). What is the original message? 

Economic Mobility; The relative income of a child (low, medium, or high) generally depends on the relative income of the child’s parents. The matrix P, given by        

$$\begin{gathered} Parent\text{'}s income \\ L M H \\ P=\left[\begin{array}{l}0.4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.1\\ 0.5\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.6\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.5\\ 0.1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0.4\end{array}\right]\begin{array}{r}\mathrm{L}\\ \mathrm{M}\\ \mathrm{H}\end{array} Child\text{'}s income \end{gathered}$$

is called a left stochastic transition matrix. For example, the entry $P_{21}=0.5$ means that $50\%$ of the children of low relative income parents will transition to the medium level of income. The diagonal entry $P_{ii}$ represents the percent of children who remain in the same income level as their parents. Assuming that the transition matrix is valid from one generation to the next, compute and interpret $P^2$. 

Consider the $2$ by $2$ squre matrix

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

If $D=ad-bc\ne 0$, show that A is nonsingular and that

$A^{-1}=\frac{1}{D}\left[\begin{array}{r}d\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-b\\ -c\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}a\end{array}\right]$

Create a situation different from any found in the text that can be represented by a matrix 

Explain why the number of columns in matrix A must equal the number of rows in matrix B when finding the product AB. 

If $a, b$, and $c\ne 0$ are real numbers with $ac = bc$, then $a = b$. Does this same property hold for matrices? In other words, if $A$, $B$, and $C$, are matrices and $AC = BC$, must $A = B$?

What is the solution of the system of equations $AX = 0$, if $A^{-1}$ exists? Discuss the solution of $AX = 0$ if $A^{-1}$ does not exist. 

True or False The equation ${(x-1)}^2-1=x(x-2)$ is an example of an identity 

True or False The rational expression $\frac{5x^2-1}{x^3+1}$ is proper. 

Factor completely: $3x^4+6x^3+3x^2$

True or False Every polynomial with real numbers as coefficients can be factored into products of linear and/or irreducible quadratic factors. 

In Problems 5–12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression 

$\text{ 5. }\frac{x}{x^2-1}$

In Problems 5–12, tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression 

$6. \frac{5x+2}{x^3-1}$

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{3x^2-2}{x^2-1}$.

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{5x^3+2x-1}{x^2-4}$.

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{3x^4+x^2-2}{x^3+8}$.

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{x(x-1)}{(x+4)(x-3)}$.

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{2x(x^2+4)}{x^2+1}$.

Write the partial fraction decomposition of each rational expression:

$\frac{4}{x(x-1)}$.

Write the partial fraction decomposition of each rational expression:

$\frac{3x}{(x+2)(x-1)}$.

Write the partial fraction decomposition of each rational expression:

$\frac{1}{x(x^2+1)}$.

Write the partial fraction decomposition of each rational expression.

$\frac{1}{(x+1)(x^2+4)}$.