2.2 Matrices: Homework

INTRODUCTION TO MATRICES

A vendor sells hot dogs and corn dogs at three different locations. His total sales (in hundreds) for January and February from the three locations are given in the table below.

Table 2.6

Place I

Place II

Place III

Represent these tables as $3 times 2$ matrices $J$ and $F$ , and answer,,, and. problems 1 - 4.

Determine total sales for the two months, that is, find $J + F$ .

$open bracket 18, 15; 14, 13; 12, 9 close bracket$

Find the difference in sales, $J - F$ .

If hot dogs sell for $3 and corn dogs for $2, find the revenue from the sale of hot dogs and corn dogs. Hint: Let $P$ be a $2 times 1$ matrix. Find $(J + F) P$ .

$open bracket 84; 68; 54 close bracket$

If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold in March. Hint: Let $R$ be a $1 times 3$ matrix with entries 1.10, 1.15, and 1.20. Find $RF$ .

Determine the sums and products in the next 5 problems. Given the matrices $A$ , $B$ , $C$ , and $D$ as follows:

$A equals open bracket 3, 6, 1; 0, 1, 3; 2, 4, 1 close bracket$ $B = [\begin{matrix} 1 & - 1 & 2 \\ 1 & 4 & 2 \\ 3 & 1 & 1 \end{matrix}]$ $C equals open bracket 1; 2; 3 close bracket$ $D equals open bracket 2, 3, 2 close bracket$

$3A minus 2B$

$[\begin{matrix} 7 & 20 & - 1 \\ - 2 & - 5 & 5 \\ 0 & 10 & 1 \end{matrix}]$

$AB + BA$

$A^{2}$

$open bracket 11, 28, 22; 6, 13, 6; 8, 20, 15 close bracket$

$2 BC$

$2 CD plus 3 AB$

$open bracket 40, 72, 61; 38, 33, 23; 39, 63, 51 close bracket$

$A^{2} B$

Let $E = [\begin{matrix} m & n \\ p & q \end{matrix}]$ and $F = [\begin{matrix} a & b \\ c & d \end{matrix}]$ , find $EF$ .

$[\begin{matrix} ma + nc & mb + nd \\ pa + qc & pb + qd \end{matrix}]$

Let $G = [\begin{matrix} 3 & 6 & 1 \\ 0 & 1 & 3 \\ 2 & 4 & 1 \end{matrix}] H = [\begin{matrix} x \\ y \\ z \end{matrix}]$ , find $GH$ .

Express the following systems as $AX = B$ , where $A$ , $X$ , and $B$ are matrices.

$4x - 5y = 6$

$5x - 6y = 7$

$[\begin{matrix} 4 & - 5 \\ 5 & - 6 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 6 \\ 7 \end{matrix}]$

$x - 2y + 2z = 3$

$x - 3y + 4z = 7$

$x - 2y - 3z = - 12$

$2x + 3z = 17$

$3x - 2y = 10$

$5y + 2z = 11$

$[\begin{matrix} 2 & 0 & 3 \\ 3 & - 2 & 0 \\ 0 & 5 & 2 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 17 \\ 10 \\ 11 \end{matrix}]$

$\begin{matrix} x & + & 2y & + & 3z & + & 2w & = & 14 \\ x & - & 2y & - & z & = & - 5 \\ y & - & 2z & + & 4w & = & 9 \\ x & + & 3z & + & 3w & = & 15 \end{matrix}$

SYSTEMS OF LINEAR EQUATIONS

Solve the following by the Gauss-Jordan Method. Show all work.

$x + 3y = 1$

$2x - 5y = 13$

(4,-1)

$x - y - z = - 1$

$x - 3y + 2z = 7$

$2x - y + z = 3$

$x + 2y + 3z = 9$

$3x + 4y + z = 5$

$2x - y + 2z = 11$

(2, -1, 3)

$x + 2y = 0$

$y + z = 3$

$x + 3z = 14$

Two apples and four bananas cost $2.00 and three apples and five bananas cost $2.70. Find the price of each.

(0.4, 0.3)

A bowl of corn flakes, a cup of milk, and an egg provide 16 grams of protein. A cup of milk and two eggs provide 21 grams of protein, and two bowls of corn flakes with two cups of milk provide 16 grams of protein. How much protein is provided by one unit of each of these three foods.

$x + 2y = 10$

$y + z = 5$

$z + w = 3$

$x + w = 5$

(4, 3, 2, 1)

$x + w = 6$

$2x + y + w = 16$

$x - 2z = 0$

$z + w = 5$

SYSTEMS OF LINEAR EQUATIONS – SPECIAL CASES

Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.

$2x + 6y = 8$

$x + 3y = 4$

$open parenthesis 4 minus 3t, t close parenthesis$

The sum of the digits of a two digit number is 9. The sum of the number and the number obtained by interchanging the digits is 99. Find the number.

$2x - y = 10$

$- 4x + 2y = 15$

Inconsistent system, no solution

$x + y + z = 6$

$3x + 2y + z = 14$

$4x + 3y + 2z = 20$

$x + 2y - 4z = 1$

$2x - 3y + 8z = 9$

$(3 - 4 / 7t, - 1 + 16 / 7t, t)$

Jessica has a collection of 15 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all three solutions.

The latest reports indicate that there are altogether 20,000 security, logistics, and medical personnel deployed on a multinational mission. The sum of the number of medical personnel and twice the security personnel equals 10,000. Furthermore, there are 5,000 more security personnel than logistics personnel. Are these reports consistent?

No, they are not consistent.

$x + y + 2z = 0$

$x + 2y + z = 0$

$2x + 3y + 3z = 0$

Find three solutions to the following system of equations.

$x + 2y + z = 12$

$y equals 3$

(5, 3, 1), (4, 3, 2) (3, 3, 3)

For what values of $k$ the following system of equations have a). No solution? b). Infinitely many solutions?

$x + 2y = 5$

$2x + 4y = k$

$x + 3y - z = 5$

( $5 - 3s + t$ , $s$ , $t$ )

Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.

INVERSE MATRICES

In the next two problems, verify that the given matrices are inverses of each other.

$[\begin{matrix} 7 & 3 \\ 2 & 1 \end{matrix}] [\begin{matrix} 1 & - 3 \\ - 2 & 7 \end{matrix}]$

$[\begin{matrix} 1 & - 1 & 0 \\ 1 & 0 & - 1 \\ 2 & 3 & - 4 \end{matrix}] [\begin{matrix} 3 & - 4 & 1 \\ 2 & - 4 & 1 \\ 3 & - 5 & 1 \end{matrix}]$

In the following problems, find the inverse of each matrix by the row-reduction method.

$[\begin{matrix} 3 & - 5 \\ - 1 & 2 \end{matrix}]$

$open bracket 2, 5; 1, 3 close bracket$

$open bracket 1, 0, 2; 0, 1, 4; 0, 0, 1 close bracket$

$open bracket 1, 1, minus 1; 1, 0, 1; 2, 1, 1 close bracket$

$open bracket 1, 2, –1; –1, –3, 2; –1, –1, 1 close bracket$

$open bracket 1, 1, 1; 3, 1, 0; 1, 1, 2 close bracket$

In the following problems, first express the system as AX = B, and then solve using matrix inverses found in the preceding four problems.

$3x - 5y = 2$

$- x + 2y = 0$

(4, 2)

$\begin{matrix} x & + & 2z & = & 8 \\ y & + & 4z & = & 8 \\ z & = & 3 \end{matrix}$

$\begin{matrix} x & + & y & - & z & = & 2 \\ x & + & z & = & 7 \\ 2x & + & y & + & z & = & 13 \end{matrix}$

(3, 3, 4)

$\begin{matrix} x & + & y & + & z & = & 2 \\ 3x & + & y & = & 7 \\ x & + & y & + & 2z & = & 3 \end{matrix}$

Why is it necessary that a matrix be a square matrix for its inverse to exist? Explain by relating the matrix to a system of equations.

If a matrix $M$ has an inverse, then the system of linear equations that has $M$ as its coefficient matrix has a unique solution. If a system of linear equations has a unique solution, then the number of equations must be the same as the number of variables. Therefore, the matrix that represents its coefficient matrix must be a square matrix.

Suppose we are solving a system $AX = B$ by the matrix inverse method, but discover $A$ has no inverse. How else can we solve this system? What can be said about the solutions of this system?

APPLICATION OF MATRICES IN CRYPTOGRAPHY

In the following problems, the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27.

**Table 2.7**
A	B	C	D	E	F	G	H	I	J	K	L	M
1	2	3	4	5	6	7	8	9	10	11	12	13
N	O	P	Q	R	S	T	U	V	W	X	Y	Z
14	15	16	17	18	19	20	21	22	23	24	25	26

In the next two problems, use the matrix A, given below, to encode the given messages.

$A equals open bracket 3, 2; 1, 1 close bracket$

In the two problems following, decode the messages that were encoded using matrix A.

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

Encode the message: WATCH OUT!

$open bracket 71; 24 close bracket open bracket 66; 23 close bracket open bracket 78; 35 close bracket open bracket 87; 36 close bracket open bracket 114; 47 close bracket$

Encode the message: HELP IS ON THE WAY.

Decode the following message:

64 23 102 41 82 32 97 35 71 28 69 32

RETURN HOME

Decode the following message:

105 40 117 48 39 19 69 32 72 27 37 15 114 47

In the next two problems, use the matrix B, given below, to encode the given messages.

$B = [\begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 2 \\ 1 & 0 & - 1 \end{matrix}]$

In the two problems following, decode the messages that were encoded using matrix $B$ .

Make sure to consider the spaces between words, but ignore all punctuation. Add a final space(s) if necessary.

Encode the message using matrix $B$ :

LUCK IS ON YOUR SIDE.

$[\begin{matrix} 12 \\ 51 \\ 9 \end{matrix}] [\begin{matrix} 11 \\ 67 \\ 2 \end{matrix}] [\begin{matrix} 19 \\ 95 \\ 14 \end{matrix}] [\begin{matrix} 14 \\ 105 \\ - 11 \end{matrix}] [\begin{matrix} 15 \\ 87 \\ - 3 \end{matrix}] [\begin{matrix} 27 \\ 91 \\ 18 \end{matrix}] [\begin{matrix} 4 \\ 67 \\ - 23 \end{matrix}]$

Encode the message using matrix $B$ :

MAY THE FORCE BE WITH YOU.

Decode the following message that was encoded using matrix B:

8 23 7 4 47 –2 15 102 –12 20 58 15 27 80 18 12 74 –7

HEAD FOR THE HILLS

Decode the following message that was encoded using matrix B:

12 69 –3 11 53 9 5 46 –10 18 95 –9 25 107 4 27 76 22 1 72 –26

APPLICATIONS – LEONTIEF MODELS

Solve the following homogeneous system.

$x + y + z = 0$

$3x + 2y + z = 0$

$4x + 3y + 2z = 0$

( $t$ , $minus 2t$ , $t$ )

Solve the following homogeneous system.

$x - y - z = 0$

$x - 3y + 2z = 0$

$2x - 4y + z = 0$

Chris and Ed decide to help each other by doing repairs on each others houses. Chris is a carpenter, and Ed is an electrician. Chris does carpentry work on his house as well as on Ed's house. Similarly, Ed does electrical repairs on his house and on Chris' house. When they are all finished they realize that Chris spent 60% of his time on his own house, and 40% of his time on Ed's house. On the other hand Ed spent half of his time on his house and half on Chris's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

$Chris equals $ 1250$ , $Ed equals $ 1, 000$

Chris, Ed, and Paul decide to help each other by doing repairs on each others houses. Chris is a carpenter, Ed is an electrician, and Paul is a plumber. Each does work on his own house as well as on the others houses. When they are all finished they realize that Chris spent 30% of his time on his own house, 40% of his time on Ed's house, and 30% on Paul's house. Ed spent half of his time on his own house, 30% on Chris' house, and remaining on Paul's house. Paul spent 40% of the time on his own house, 40% on Chris' house, and 20% on Ed's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

Given the internal consumption matrix $A$ , and the external demand matrix $D$ as follows.

$A = [\begin{matrix} . 30 & . 20 & . 10 \\ . 20 & . 10 & . 30 \\ . 10 & . 20 & . 30 \end{matrix}] D = [\begin{matrix} 100 \\ 150 \\ 200 \end{matrix}]$

Solve the system using the open model: $X = AX + D$ or $X = {(I - A)}^{- 1} D$

$open bracket 315.34; 383.52; 440.34 close bracket$

Given the internal consumption matrix $A$ , and the external demand matrix $D$ as follows.

$A = [\begin{matrix} . 05 & . 10 & . 10 \\ . 10 & . 15 & . 05 \\ . 05 & . 20 & . 20 \end{matrix}] D = [\begin{matrix} 50 \\ 100 \\ 80 \end{matrix}]$

Solve the system using the open model: $X = AX + D$ or $X = {(I - A)}^{- 1} D$

An economy has two industries, farming and building. For every $1 of food produced, the farmer uses $.20 and the builder uses $.15. For every $1 worth of building, the builder uses $.25 and the farmer uses $.20. If the external demand for food is $100,000, and for building $200,000, what should be the total production for each industry in dollars?

$Farming = $ 201, 754.38, Building = $ 307, 017.54$

An economy has three industries, farming, building, and clothing. For every $1 of food produced, the farmer uses $.20, the builder uses $.15, and the tailor $.05. For every $1 worth of building, the builder uses $.25, the farmer uses $.20, and the tailor $.10. For every $1 worth of clothing, the tailor uses $.10, the builder uses $.20, the farmer uses $.15. If the external demand for food is $100 million, for building $200 million, and for clothing $300 million, what should be the total production for each in dollars?

Suppose an economy consists of three industries $F$ , $C$ , and $T$ . The following table gives information about the internal use of each industry's production and external demand in dollars.

**Table 2.8**
	$F$	$C$	$T$	Demand	Total
$F$	30	10	20	40	100
$C$	20	30	20	50	120
$T$	10	10	30	60	110

Find the proportion of the amounts consumed by each of the industries; that is, find the matrix $A$ .

$open bracket 30 / 100, 10 / 120, 20 / 110; 20 / 100, 30 / 120, 20 / 110; 10 / 100, 10 / 120, 30 / 110 close bracket$

If in the preceding problem, the consumer demand for $F$ , $C$ , and $T$ becomes 60, 80, and 100, respectively, find the total output and the internal use by each industry to meet that demand.

CHAPTER REVIEW

To reinforce her diet, Mrs. Tam bought a bottle containing 30 tablets of Supplement $A$ and a bottle containing 50 tablets of Supplement $B$ . Each tablet of supplement $A$ contains 1000 mg of calcium, 400 mg of magnesium, and 15 mg of zinc, and each tablet of supplement $B$ contains 800 mg of calcium, 500 mg of magnesium, and 20 mg of zinc.

Represent the amount of calcium, magnesium and zinc in each tablet as a $2 times 3$ matrix.
Represent the number of tablets in each bottle as a row matrix.
Use matrix multiplication to determine the total amount of calcium, magnesium, and zinc in both bottles.

$open bracket 1000, 400, 15; 800, 500, 20 close bracket$
$open bracket 30, 50 close bracket$

Let matrix $A = [\begin{matrix} 1 & - 1 & 3 \\ 3 & - 2 & 1 \end{matrix}]$ and $B = [\begin{matrix} 3 & 3 & - 1 \\ 1 & 4 & - 3 \end{matrix}]$ . Find the following.

$\frac{1}{2} (A + B)$
$3A minus 2B$

$open bracket 2, 1, 1; 2, 1, minus 1 close bracket$
$[\begin{matrix} - 3 & - 9 & 11 \\ 7 & - 14 & 9 \end{matrix}]$

Let matrix $C = [\begin{matrix} 1 & 1 & - 1 \\ 2 & 1 & 1 \\ 1 & 0 & 1 \end{matrix}]$ and $D = [\begin{matrix} 2 & - 3 & - 1 \\ 3 & - 1 & - 2 \\ 3 & - 3 & - 2 \end{matrix}]$ . Find the following.

$2 open parenthesis C minus D close parenthesis$
$C minus 3D$

$[\begin{matrix} - 2 & 8 & 0 \\ - 2 & 4 & 6 \\ - 4 & 6 & 6 \end{matrix}]$
$[\begin{matrix} - 5 & 10 & 2 \\ - 7 & 4 & 7 \\ - 8 & 9 & 7 \end{matrix}]$

Let matrix $E = [\begin{matrix} 1 & - 1 \\ 2 & 3 \\ 1 & 2 \end{matrix}]$ and $F = [\begin{matrix} 2 & 1 & - 1 \\ 1 & 2 & - 3 \end{matrix}]$ . Find the following.

$2 EF$
$3 FE$

$[\begin{matrix} 2 & - 2 & 4 \\ 14 & 16 & - 22 \\ 8 & 10 & - 14 \end{matrix}]$
$[\begin{matrix} 9 & - 3 \\ 6 & - 3 \end{matrix}]$

Let matrix $G = [\begin{matrix} 1 & - 1 & 3 \\ 3 & 2 & 1 \end{matrix}]$ and $H = [\begin{matrix} a & b \\ c & d \\ e & f \end{matrix}]$ . Find the following.

$2 GH$
$HG$

$[\begin{matrix} 2a - 2c + 6 e & 2b - 2d + 6f \\ 6a + 4c + 2 e & 6b + 4d + 2f \end{matrix}]$
$[\begin{matrix} a + 3b & - a + 2b & 3a + b \\ c + 3d & - c + 2d & 3c + d \\ e + 3f & - e + 2f & 3e + f \end{matrix}]$

Solve the following systems using the Gauss-Jordan Method.

$\begin{matrix} x & + & 3y & - & 2z & = & 7 \\ 2x & + & 7y & - & 5z & = & 1 \\ x & + & 5y & - & 3z & = & 1 \end{matrix}$
$\begin{matrix} 2x & - & 4y & + & 4z & = & 2 \\ 2x & + & y & + & 9z & = & 1 \\ 3x & - & 2y & + & 2z & = & 7 \end{matrix}$

(2, 1, –1)
(3, 2, 1)

An apple, a banana and three oranges or two apples, two bananas, and an orange, or four bananas and two oranges cost $2. Find the price of each.

Apple = $.50; banana = $.30; orange = $.40

Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then determine one particular solution.

$\begin{matrix} x & + & y & + & z & = & 6 \\ 2x & - & 3y & + & 2z & = & 1 \\ 3x & - & 2y & + & 3z & = & 1 \end{matrix}$
$\begin{matrix} x & + & y & + & 3z & = & 4 \\ x & + & z & = & 1 \\ 2x & - & y & = & 2 \end{matrix}$

$x = 6 - t$ , $y equals 0$ , $z = t$ ; (5, 0, 1)
no solution

Elise has a collection of 12 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all possible solutions.

$n = 3t - 12$ , $d = - 4t + 24$ , $q = t$ ; $n equals 3$ , $d equals 4$ , $q equals 5$

Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then find a particular solution.

$\begin{matrix} x & + & 2y & = & 4 \\ 2x & + & 4y & = & 8 \\ 3x & + & 6y & - & 3z & = & 3 \end{matrix}$
$\begin{matrix} x - 2y + 2z = 1 \\ 2x - 3y + 5z = 4 \end{matrix}$

$x = 4 - 2t$ , $y = t$ , $z equals 3$ ; (4, 0, 3)
$x = 5 - 4t$ , $y = 2 - t$ , $z = t$ ; (1, 1, 1)

Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then provide one particular solution.

$\begin{matrix} 2x & + & y & - & 2z & = & 0 \\ 2x & + & 2y & - & 3z & = & 0 \\ 6x & + & 4y & - & 7z & = & 0 \end{matrix}$
$\begin{matrix} 3x & + & 4y & - & 3z & = & 5 \\ 2x & + & 3y & - & z & = & 4 \\ x & + & 2y & + & z & = & 1 \end{matrix}$

$x equals .5t$ , $y = t$ , $z minus 2t$ ; (1, 2, 2)
no solution

Find the inverse of the following matrices.

$open bracket 2, 3; 3, 5 close bracket$
$open bracket 1, 1, 1; 1, 2, 1; 2, 3, 1 close bracket$

$[\begin{matrix} 5 & - 3 \\ - 3 & 2 \end{matrix}]$
$[\begin{matrix} 1 & - 2 & 1 \\ - 1 & 1 & 0 \\ 1 & 1 & - 1 \end{matrix}]$

Solve the following systems using the matrix inverse method.

$\begin{matrix} 2x & + & 3y & + & z & = & 1 \\ x & + & 2y & + & z & = & 9 \\ x & + & y & + & z & = & 5 \end{matrix}$
$\begin{matrix} x & + & 2y & - & 3z & + & w & = & 7 \\ x & - & z & = & 4 \\ x & - & 2y & + & z & = & 0 \\ y & - & 2z & + & w & = & - \end{matrix}$

(-1, 4, 2)
(6, 4, 2, -1)

Use matrix $A$ , given below, to encode the following messages. The space between the letters is represented by the number 27, and all punctuation is ignored.

$A equals open bracket 1, 2, 0; 1, 2, 1; 0, 1, 0 close bracket$

TAKE IT AND RUN.
GET OUT QUICK.

$open bracket 22; 33; 1 close bracket open bracket 59; 68; 27 close bracket open bracket 74; 75; 27 close bracket open bracket 22; 49; 4 close bracket open bracket 60; 74; 21 close bracket$
$open bracket 17; 37; 5 close bracket open bracket 57; 78; 15 close bracket open bracket 74; 91; 27 close bracket open bracket 39; 42; 9 close bracket open bracket 65; 92; 27 close bracket$

Decode the following messages that were encoded using matrix $A$ in the above problem.

44, 71, 15, 18, 27, 1, 68, 82, 27, 69, 76, 27, 19, 33, 9
37, 64, 15, 36, 54, 15, 67, 75, 20, 59, 66, 27, 39, 43, 12

NO PAIN NO GAIN
GO FOR THE GOLD

Chris, Bob, and Matt decide to help each other study during the final exams. Chris's favorite subject is chemistry, Bob loves biology, and Matt knows his math. Each studies his own subject as well as helps the others learn their subjects. After the finals, they realize that Chris spent 40% of his time studying his own subject chemistry, 30% of his time helping Bob learn chemistry, and 30% of the time helping Matt learn chemistry. Bob spent 30% of his time studying his own subject biology, 30% of his time helping Chris learn biology, and 40% of the time helping Matt learn biology. Matt spent 20% of his time studying his own subject math, 40% of his time helping Chris learn math, and 40% of the time helping Bob learn math. If they originally agreed that each should work about 33 hours, how long did each work?

$x = 40 / 33 t, y = 36 / 33 t, z = t; Chris = 40 hrs, Bob = 36hrs, Matt = 33hrs$

As in the previous problem, Chris, Bob, and Matt decide to not only help each other study during the final exams, but also tutor others to make a little money. Chris spends 30% of his time studying chemistry, 15% of his time helping Bob with chemistry, and 25% helping Matt with chemistry. Bob spends 25% of his time studying biology, 15% helping Chris with biology, and 30% helping Matt. Similarly, Matt spends 20% of his time on his own math, 20% helping Chris, and 20% helping Bob. If they spend respectively, 12, 12, and 10 hours tutoring others, how many total hours are they going to end up working?

$Chris = 34 . 1hrs, Bob = 32 . 2hrs, Matt = 35 . 2hrs$

Adapted from Applied Finite Mathematics by Rupinder Sekhon (De Anza College), originally published by OpenStax CNX (cnx.org, collection col10613), licensed under CC BY 3.0. Changes were made. License: CC-BY-3.0.