For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse.
a. E = 1 0 3
0 1 0
0 0 1
b. E = 0 0 1
0 1 0
1 0 0
c. E = 1 0 0
0 1/2 0
0 0 1
d. E = 1 0 0
-2 1 0
0 0 1
e. E = 0 1 0
1 0 0
0 0 1
f. E = 1 0 0
0 1 0
0 0 5

Answer:

Step-by-step explanation:

e

An expression equivalent to (a - b)³ is a³ - 3a²b + 3ab² - b³.

The fractions 4/10, 6/15, 8/20, etc. are identical to 2/5. In the reduced form, equivalent fractions have the same value. Explanation: When writing equivalent fractions, the numerator and denominator should be multiplied or divided by the same number.

One way to expand (a - b)³ is to use the binomial formula:

(a - b)³ = C(3,0) * a³ * (-b)^0 + C(3,1) * a² * (-b) + C(3,2) * a * (-b)² + C(3,3) * a * (-b)³

where C(n,k) denotes the number of ways there are to select k objects from a set of n objects, and "n choose k" is the binomial coefficient.

Simplifying the above expression, we get:

(a-b)³ = a³-3a²b+3ab²-b³.

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Answer: There were 52 Republicans and 46 Democrats.

Step-by-step explanation:

Let's represent the number of Republicans as "x" and the number of Democrats as "y".

From the problem, we know that:

x + y = 98 (since the total number of senators is 98)

y = x - 6 (since there were 6 fewer Democrats than Republicans)

We can substitute the second equation into the first equation to get:

x + (x-6) = 98

2x - 6 = 98

2x = 104

x = 52

Now that we know that there were 52 Republicans, we can substitute that into the second equation to find the number of Democrats:

y = x - 6

y = 52 - 6

y = 46

Therefore, there were 52 Republicans and 46 Democrats.

Answer:

There were 52 Republicans and 46 Democrats.

Step-by-step explanation:

Let's make a system of equations to represent this! D = democrats and R = republicans.

[tex]d + 6 = r[/tex]

[tex]r + d = 98[/tex]

Now let's set them both equal to d.

To do this for the first equation, subtract 6 from both sides to get:

[tex]d =r - 6[/tex]

To do this for the second equation, subtract r from both sides to get:

[tex]d = 98 - r[/tex]

Now, let's set them equal to eachother.

[tex]r - 6 = 98 - r[/tex]

Add r to both sides

[tex]2r -6 = 98[/tex]

Add 6 to both sides

[tex]2r = 104[/tex]

Divide both sides by two

[tex]r = 52[/tex]

There were 52 republicans. Now let's sub 52 in for r in one of the equations.

[tex]d + 6 = 52[/tex]

subtract six from both sides

[tex]d = 46[/tex]

There were 46 democrats.

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The ability of the normal distribution to approximate the binomial distribution under appropriate conditions justifies the use of the normal distribution for the sampling distribution of the proportion.

A sampling distribution is a statistical probability distribution derived from a larger number of samples gathered from a certain population. The sampling distribution of a particular population is the distribution of frequencies of a range of possible outcomes for a population statistic.

A population is the whole pool from which a statistical sample is selected in statistics. A population can be defined as a large group of people, things, events, medical visits, or measures. A population can thus be defined as an aggregate observation of persons linked by a common trait.

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The computer response time is a crucial application of the gamma and exponential distribution. The proportion of computers studied that will have a response time less than 3 seconds is 0.6321 or 63.21%.

If a study of 200 computers shows that the response time in seconds has an exponential distribution with a mean of 3 seconds, then the probability that the response time exceeds 5 seconds and the proportion of the computers studied will have a response time less than 3 seconds can be calculated as follows:The mean of the exponential distribution is equal to the reciprocal of the rate parameter, λ, i.e.,E(X) = λ-1 Given, the mean response time, E(X) = 3 seconds.The mean of the exponential distribution, E(X) = λ-1= 3 seconds. Therefore,λ = 1/3. The probability that the response time exceeds 5 seconds can be calculated as:P(X > 5) = e -λx= e -(1/3) (5)= 0.0498 The probability that the response time exceeds 5 seconds is 0.0498 or 4.98%.The proportion of computers studied that will have a response time less than 3 seconds can be calculated as:P(X < 3) = 1 - P(X > 3)= 1 - e -λx= 1 - e -(1/3) (3)= 1 - e -1= 1 - 0.3679= 0.6321 or 63.21%.

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The cost of 9 gallons of gas is 24.75$

A unitary method is a mathematical technique for finding the value of a single unit and then multiplying it by any number of other units to derive the value of the provided units. Employing the unitary method, we can solve the above problem.

Given that,

The cost of 8 gallons of gas =  22$

The cost of 1 gallon of gas = 22 ÷ 8

                                            = 2.75$

Therefore, the cost of  9 gallons of gas = 2.75 × 9

                                                                 = 24.75$.

Hence, the cost of 9 gallons of gas is 24.75$.

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As per the volume, the dimension of an Extra Large Box is 3.78 feet.

Let's call the length of one side of the cube "s". Since the volume of the cube is given as 512 cubic feet, we can set up an equation to relate the volume to the length of one side:

Volume of cube = s³ = 512 cubic feet

To solve for "s", we can take the cube root of both sides of the equation:

s = ∛512

We can simplify this expression by finding the prime factorization of 512:

512 = 2⁹

Therefore, we can rewrite the expression for "s" as:

s = ∛2⁹

Using the properties of exponents, we know that the cube root of 2^9 is the same as 2 raised to the power of (1/3) times 9:

s = [tex]2^{1/3} \times 9^{1/3}[/tex]

We can simplify this expression further by recognizing that 9 is a perfect cube, and its cube root is 3:

s = [tex]2^{1/3} \times 3[/tex]

Therefore, the length of one side of the cube-shaped box is:

s = [tex]2^{1/3} \times 3[/tex] feet

Since all sides of the cube are equal in length, the dimensions of the box are:

Length = Width = Height = [tex]2^{1/3} \times 3[/tex] feet = 3.78 feet.

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Complete Question:

Head Stevedore loads Extra Large Boxes, also in the shape of perfect cubes. The volume of each box is 512 cubic feet. What are the dimension of an Extra Large Box?

The fixed amount Hester earns for each television is $65.

A straight line in a two-dimensional coordinate system is described by a linear equation. Using m as the line's slope and b as the y-intercept, it has the generic form y = mx + b.

Mathematical applications of linear equations include algebra, geometry, and calculus. They are employed to represent and examine interactions between different variables, such as the connection between time and space in a straightforward linear motion issue.

Let us suppose the fixed amount = x.

Then, for 15 televisions she earns:

15*$25 = $375

Now, she earned a total of 1350, thus:

15x + $375 = $1,350

15x = $975

x = $65

Therefore, the fixed amount Hester earns for each television is $65.

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If the two removed squares have the same color, then it is possible to cover the truncated checkerboard with non-overlapping dominoes. If they have different colors, then it will be impossible to cover the truncated checkerboard with non-overlapping dominoes.

To prove that the chess board can be completely covered by non-overlapping dominoes, where each domino covers exactly two squares, we can color the board alternatively in black and white. The two colors will make the board have 32 black squares and 32 white squares. Each domino will cover one white and one black square. Since there are the same number of black and white squares, there is an equal number of squares that each domino can cover so it is possible to cover the entire board with dominoes without overlap.

To determine whether the cut checkerboard can be covered by non-overlapping dominoes, note that we have 62 squares, 31 of which are black, and 31 white. This means that one color will have one extra square, in this case black. Therefore, if the two removed squares are of different colors, it will be impossible to cover the cut checkerboard with non-overlapping dominoes.However, if the two removed squares have the same color, then it will be possible to cover the checkerboard with non-overlapping dominoes. This is because we will still have an equal number of black and white squares even after the removal of the two squares.The same logic applies to the truncated checkerboard.

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Serena leans a 24-foot ladder against a 22.8-foot-tall building to climb on the roof is 71.8 degree.

Measurement of Angle:

The measurement of angles is done using basic geometric tools such as protractors and compasses. This tool helps to find precise measurements of angles. A protractor helps provide accurate measurements of angles, while compasses help construct angles. Angles are measured in three ways: degrees, radians, and revolutions.

According to the Question:

Suppose the measure of angle is θ, then we have

   Sinθ = 22.8/24

⇒ Sinθ = 0.95

Therefore,

θ = Sin⁻¹ (0.95)

  = 71.8 degree.

Therefore, Serena leans a 24-foot ladder against a 22.8-foot-tall building to climb on the roof is 71.8 degree.

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The answer is 12units

Answer: y is equal to 8

Step-by-step explanation:

by substituting the x for its vale of three we can add the two values to get 8 or y=8

Answer:

235.6 cm²

Step-by-step explanation:

The formula to find the area of a sector is,

[tex]\sf A=\frac{\theta}{360}*\pi r^2[/tex]

Here,

Θ = angle

r = radius

Let us find it now.

[tex]\sf A=\frac{\theta}{360}*\pi r^2\\\\\sf A=\frac{270}{360}*\pi *10*10\\\\\sf A=\frac{270}{360}*\pi *100\\\\A = 235.61 cm^2[/tex]

Answer:

20%

Step-by-step explanation:0% of 4445=

In the context of this problem, if we reject the null hypothesis, we can conclude that the new design is more effective in terms of accuracy. If we fail to reject the null hypothesis, we can't conclude anything about the effectiveness of the new design

When given the resulting p-values, we reject the null hypotheses (assuming a 0.05 significance level) if the p-value is less than 0.05. We fail to reject the null hypotheses if the p-value is greater than or equal to 0.05.If we reject the null hypothesis, it means that there is strong evidence against it, and we can say that the alternative hypothesis is supported by the data. If we fail to reject the null hypothesis, it means that there is insufficient evidence to reject it, and we can't say that the alternative hypothesis is supported by the data.

In the context of this problem, if we reject the null hypothesis, we can conclude that the new design is more effective in terms of accuracy. If we fail to reject the null hypothesis, we can't conclude anything about the effectiveness of the new design.In either case, whether we reject or fail to reject the null hypothesis, we wouldn't proceed with changing the design of the arrow based solely on the statistical analysis. We would need to consider other factors such as cost, feasibility, and practicality before making any decisions about changing the design.

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Answer:

for the tringle:

area=[tex]p^{2} -2p-3[/tex]

for the frame:

area=[tex]8x+28[/tex]

Step-by-step explanation:

For the tringle:-

Area=1/2bh

Area=1/2(2p-6)(p+1)

Area=1/2([tex]2p^{2} -4p-6[/tex])

Area=[tex]p^{2} -2p-3[/tex]

For the picture frame:-

part 1:the area of the big frame

Area=lw

Area=(x+7)(x+4)

Area=[tex]x^{2} +11x+28[/tex]

part 2:the area of the small frame

Area=lw

Area=(x+3)(x)

Area=[tex]x^{2} +3x[/tex]

To find the area of the shaded regions:

Area of shaded region=area of the big frame-the area of the small frame

Area of shaded region=[tex](x^{2} +11x+28)-(x^{2} +3x)[/tex]

Area of shaded region=[tex]8x+28[/tex]

Answer:

No solution

Step-by-step explanation:

They have the same slope, which makes them parallel to each other. The y-intercepts are different so they do not overlap. If they overlap, then you know they have either one or infinite solutions. They do not so the answer is no solution.

Answer:

Step-by-step explanation:

First, we need to find the coordinates of the center of the circle. Since the equation of the circle is x^2 + y^2 = 40, we have x^2 = 40 - y^2. Substituting this into the equation of the tangent line I, we get:

(40 - y^2) + y(2x) = 40

2xy - y^2 = 0

y(2x - y) = 0

Since the line I crosses the x-axis at point P, we have y = 0 at that point. Therefore, 2x - y = 0, and we can solve for x:

2x = y

2x = 6

x = 3

So the center of the circle is at the point (3, 0).

Next, we need to find the radius of the circle. Since A is on the circle, we have:

3^2 + 6^2 = r^2

r^2 = 45

r = sqrt(45) = 3sqrt(5)

Now we can find the coordinates of point O, which is the center of the circle:

O = (3, -3sqrt(5))

Finally, we can find the area of triangle OAP:

OA = sqrt((2-3)^2 + (6+3sqrt(5))^2) = sqrt(69)

AP = 2

Using the formula for the area of a triangle, we have:

Area(OAP) = 1/2 * OA * AP = 1/2 * sqrt(69) * 2 = sqrt(69)

Therefore, the area of triangle OAP is sqrt(69) square units.

The student is male, which is P(male).Using the rule of complementary events, P(male) = 1 - P(female)P(male) = 1 - 0.572 = 0.428Therefore, the probability that the student is male is 0.428 or 42.8%.

The complimentary event is a part of probability theory. It is the event that occurs when the event E does not occur. In other words, it is a set of outcomes in the sample space that are not included in the outcomes of event E. The notation for the complement of E is E'. Rule for Complimentary EventsThe rule for complementary events can be expressed in two ways:

P(E) = 1 - P(E)P(E) + P(E') = 1Example # 12:Let the probability that Mary can work a problem be P(E) = 0.70.We need to find the probability that Mary cannot work the problem, which is P(E').Using the rule of complementary events,P(E') = 1 - P(E)P(E') = 1 - 0.70 = 0.30Therefore, the probability that Mary cannot work the problem is 0.30 or 30%.Example # 13:Let P(female) be the probability that the student is female. We are given that P(female) = 0.572.

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Answer:

0. 25 25/100

Step-by-step explanation:

The probability of producing the genotype AABBCC in a cross AaBbCc × AaBbCc is 1/64.

A Punnett square is a grid used to forecast the likelihood of an offspring of a mating between two parents with known genotypes. The grid consists of four boxes or cells with one parent's gamete genotype listed along the top, and the other parent's gamete genotype listed down the side.

To determine what proportion of their offspring will possess a certain genotype or phenotypic characteristic, the gamete genotypes are combined using the grid.

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The equation that models the shape of the monument is simply y = 25.

The equation that models the shape of the parabolic monument can be written in the form of a quadratic function, y = ax^2 + bx + c, where y is the height of the monument at a given distance x from the center of the base.

Since the monument has a height of 25 feet at the center of the base, we know that the vertex of the parabolic shape is located at (0, 25). Also, since the base width is 30 feet, the distance from the center of the base to either side is 15 feet.

Therefore, we can use the information about the vertex and the width to write the equation as y = -a(x-15)^2 + 25.

To determine the value of a, we need another point on the curve. Let's use one of the endpoints of the base, which is (15, 0). Plugging these values into the equation, we get:

0 = -a(15-15)^2 + 25

0 = 25

This is not possible, so we need to adjust the equation to fit the known points. We can rewrite the equation as y = a(x-15)^2 + 25, and solve for a using the other endpoint of the base, which is (-15, 0):

25 = a(-15-15)^2 + 25

0 = 900a

a = 0

This means that the equation that models the shape of the monument is simply y = 25.

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To summarize, a perfectly egalitarian society has a Gini index of 0, and a perfectly totalitarian society has a Gini index of 1.

The Gini index (G) measures how much the income distribution differs from absolute equality.

On the other hand, a perfectly totalitarian society (where a single person receives all the income) will have a Gini index of 1. This means that there is a huge inequality between individuals, with the single person receiving all of the income.

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The surface area of the triangular prism is 2016 [tex]feet ^2[/tex], we get to the answer by adding area of all faces in this prism.

What is surface area ?

Surface area is the sum of the areas of all the faces or surfaces of a 3D object, measured in square units.

To find the surface area of the triangular prism, we need to calculate the area of each of its faces and then add them up.

First, let's find the area of the triangular base.

Area of triangle = (1/2) x base x height

Area of triangle = (1/2) x 18 x 24

Area of triangle = 216 [tex]feet ^2[/tex]

Next, let's find the area of each rectangular face.

Area of rectangle = length x width

Area of rectangle 1 = 24 x 22 = 528 [tex]feet ^2[/tex]

Area of rectangle 2 = 18 x 22 = 396 [tex]feet ^2[/tex]

Area of rectangle 3 =  22 x 30 = 660 [tex]feet ^2[/tex]

Now, we can add up the areas of all three faces to get the total surface area of the prism:

Surface area = area of rectangle (1+2+3) + 2 x area of triangle

Surface area = 528 + 396 + 660 + 2(216)

Surface area = 2016 [tex]feet ^2[/tex]

Therefore, the surface area of the triangular prism is 2016 [tex]feet ^2[/tex].

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Considering the Markov chain with transition matrix, the chain does have stationary distribution which exhibits State 1 is transient and the stationary distribution is (2/9, 4/9, 8/27, 16/81, 32/729).

(a) Yes, the chain has a stationary distribution. To find it, we need to solve the system of equations π = πP, where π   is the vector of probabilities for each state and P is the transition matrix. This gives us:

π_{1} = π(1/2)

π_{2} = π(1/3) + π(2/2)

π_{3}= π(2/4) + π(3/2)

π_{4} = π(3/5) + π(4/2)

π_{5}= π4/5

We also have the normalization condition π1 + π2 + π3 + π4 + π5 = 1.

Solving this system of equations, we get:

π_{1} = 10/97

π_{2} = 30/97

π_{3}= 40/97

π_{4} = 14/97

π_{5}= 3/97

So the stationary distribution is (10/97, 30/97, 40/97, 14/97, 3/97).

(b) State 1 is transient, and all other states are recurrent.

(c) Yes, the chain still has a stationary distribution. We need to solve the system of equations  π = Pπ, where P is the new transition matrix. This gives us:

π_{1} = π(1/2)

π_{2} = π(1/3) + π(2/2)

π_{3}= π(2/4) + π(3/2)

π_{4} = π(3/5) + π(4/2)

π_{5}= π4/5

We also have the normalization condition π1 + π2 + π3 + π4 + π5 = 1.

Solving this system of equations, we get:

π_{1} = 2/9

π_{2}  = 4/9

π_{3} = 8/27

π_{4} = 16/81

π_{5}  = 32/729

So the stationary distribution is (2/9, 4/9, 8/27, 16/81, 32/729)

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As a result, there are 180 students in the seventh grade.

A level οr prοgressiοn οn a scale, since abοut rank, success, excellence, wοrth, οr intensity: the highest grade οf paper. A stage οr phase in a cοurse and prοcess is a grοup οf peοple οr οbjects that have a same strength is relatively, quality, etc.

We can utilize percentage tο determine the tοtal quantity οf pupils in the class if 9 kids make up 5% οf a seventh-grade class. If we assume that there are x pupils inside the seventh grade οverall, we can calculate the prοpοrtiοn as fοllοws:

9 / x = 5 / 100

We may crοss-multiply and simplify tο find x's value:

9 × 100 = 5 × x

900 = 5x

x = 900 / 5

x = 180

In the seventh grade, there are 180 students.

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By answering the presented question, we may conclude that Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.

An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematic, and form. They are used in the representation of mathematical formulas, the solution of equations, and the simplification of mathematical relationships.

To simplify the expression,

[tex]a(b+c) = ab + ac\\9(4/3m-5-2/3m+2) = 9(4/3m - 2/3m - 5 + 2)\\= 9(2/3m - 3)\\= 6m - 27[/tex]

Therefore, the expression 9(4/3m-5-2/3m+2) is equivalent to 6m - 27.

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Answer:

£234.36

Step-by-step explanation:

The general solution of the given trignometric function tan3x . 1/tan24° - 1 = 0​ is x = nπ/3 + 2π/45.

Trigonometry is a field of mathematics that deals with correlations between angles and length ratios (from the Ancient Greek v (trgnon) "triangle" and v (métron) "measure").

The field was created in the Hellenistic era in the third century BC as a result of the use of geometry in astronomical research.

While Indian mathematicians produced the earliest tables of values for trigonometric ratios (also known as trigonometric functions), such as sine, the Greeks concentrated on chord calculation.

Trigonometry has been used historically in fields like geodesy, surveying, celestial mechanics, and navigation.

So, we have the function:

tan3x . 1/tan24° - 1 = 0​

Now, solve as follows:

tan3x . 1/tan24° - 1 = 0​

tan3x - tan24° = 0

tan3x = tan24°

tan3x = tan(24π/180)

3x = nπ + 2π/15

x = nπ/3 + 2π/45

Therefore, the general solution of the given trignometric function tan3x . 1/tan24° - 1 = 0​ is x = nπ/3 + 2π/45.

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Answer:

b = 12.12 (or) 7 sqrt 3

Step-by-step explanation:

a^2 + b^2 = c^2

7^2 + b^2 = 14^2

49 + b^2 = 196

-49 -49

b^2 = 147

b = sqrt 147

decimal form:

b = 12.12

exact form:

b = 7 sqrt 3