Total Number of Factors for N = (a+1) (b+1) (c+1) is the formula for how many factors there are in total for a given number.
what is prime factor?Any natural number other than 1 with just itself and the number 1 as its divisors is considered a prime factor. Actually, 2, 3, 5, 7, 11, and so on are some of the first prime numbers. an amount that has been multiplied to produce another amount. By dividing 15 by 3 and 5, for instance, we get 3 5 = 15. main elements: Prime factors include all factors that are prime but are not composite. A few prime factors of 30 are 2, 3, and 5. Only 2 and 3 are prime factors of 12, so listing 2 twice as 2 2 3 (or 22 3) is necessary to factorize 12. The sum of 2 + 3 is not 12.
Total Number of Factors for N = (a+1) (b+1) (c+1) is the formula for how many factors there are in total for a given number.
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what is the y value when x= -1?
what is the y value when x=0?
what is the y value when x=2?
Answer:
-1, y = 2
0, y = 3
2, y = 5
Step-by-step explanation:
-1, y = -1 + 3
0, y = 0 + 3
2, y = 2 + 3
let a be an n x n matrix with an eigenvalue of multiplicity n. show that a is diagonalizable if and only if a = i
An n x n matrix a with an eigenvalue of multiplicity n is diagonalizable if and only if a = i, where i is the identity matrix.
Suppose a is diagonalizable. Then there exists an invertible matrix P such that a = PDP^(-1), where D is a diagonal matrix. Since a has an eigenvalue of multiplicity n, the diagonal entries of D are all equal to that eigenvalue. Therefore, a = PDP^(-1) = P(lambda I)P^(-1) for some scalar lambda. But since the eigenvalue has multiplicity n, lambda must equal the eigenvalue, which implies that D = lambda I. Therefore, a = [tex]P(lambda I)P^(-1) = PDP^(-1)[/tex] = P(lambda I)P^(-1) = lambda PPP^(-1) = lambda I. Thus, a = lambda I, and since the eigenvalue has multiplicity n, we have lambda = 1. Therefore, a = i.
Conversely, suppose a = i. Then a is trivially diagonalizable, since any matrix is diagonalizable if and only if it is already diagonal. Therefore, a is diagonalizable, and the proof is complete.
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the two rectangles are similar. which is a correct proportion for corresponding sides?
A.12/8=x/8 B.12/4=x/8 C.12/4=x/20 D.4/12=x/8
The correct proportion for corresponding sides of the two similar rectangles is D. 4/12 = x/8.
To determine the correct proportion for corresponding sides, we need to compare the lengths of corresponding sides of the two rectangles. Let's denote the length of one side of the first rectangle as 12 units and the length of the corresponding side of the second rectangle as x units.
Option A states that 12/8 = x/8. However, this would imply that the length of the corresponding side in the second rectangle is equal to the length of the corresponding side in the first rectangle, which would mean the rectangles are congruent, not similar.
Option B suggests that 12/4 = x/8. By simplifying the equation, we get 3 = x/8, which implies that x = 24. This proportion does not hold since the length of the corresponding side should be less than 12 (the length of the corresponding side in the first rectangle).
Option C states that 12/4 = x/20. Simplifying this equation gives us 3 = x/20, which implies that x = 60. This proportion also does not hold since the length of the corresponding side should be less than 12 (the length of the corresponding side in the first rectangle).
Option D states that 4/12 = x/8. By simplifying the equation, we get 1/3 = x/8. This proportion holds, indicating that the length of the corresponding side in the second rectangle is one-third of the length of the corresponding side in the first rectangle. Therefore, the correct proportion for corresponding sides is D. 4/12 = x/8.
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A construction crew in lengthening a road. Let L be the total length of the road (in miles). Let D be the number of days the crew has worked. Suppose that L=2D+300 gives L as a function of D. The crew can work for at most 90 days
The given equation L = 2D + 300 represents the relationship between the total length of the road, L (in miles), and the number of days the crew has worked, D.
However, it's mentioned that the crew can work for at most 90 days. Therefore, we need to consider this restriction when determining the maximum possible length of the road.
Since D represents the number of days the crew has worked, it cannot exceed 90. We can substitute D = 90 into the equation to find the maximum length of the road:
L = 2D + 300
L = 2(90) + 300
L = 180 + 300
L = 480
Therefore, the maximum possible length of the road is 480 miles when the crew works for 90 days.
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Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
Answer:
Step-by-step explanation:
Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
THIS IS THE COMPLETE QUESTION BELOW;
1. Explain how you calculate the net force in any direction on the box.
2. Suppose an upward force of 15 N is added to the box. What will be the net vertical force on the box?
3. What force could be applied to the box to make the net force in the horizontal direction zero? Explain.
4. Suppose a force of 25 N to the right is added to the box. What will be the net force to the right?
CHECK THE ATTACHMENT FOR THE FIQURE.
1)since the given horizontal forces are acting on the box at opposite direction then
NET FORCE= (100- 50)N
= 50N( this is because the 100N and 50N acting Horizontally will cancel each other, then the remaining force is 50N.
✓Since, the two vertical forces also acted on the box in opposite direction, then
NET FORCE= (25 25)N
= 0N
Note: horizontal forces are one acting from right to left and vice versa, while vertical forces are ones acting from up to down and vice versa.
HENCE, NET FORCE = 50N + 0N= 50N
2) since the net vertical force was 0N, if
15N force is added then we still have NET FORCE OF 15N
3) if force of 50N is applied to horizontal forces moving from left to right then NET HORIZONTAL FORCES= 0N
4) since, the net force for the box is 50N, if Horizontal force of 25N is added moving left to right then NET FORCE = 25N for whole box.
can answer this please
Answer:
[tex] \frac{a}{b} = \frac{3}{2} [/tex]
Step-by-step explanation:
Since both the triangles are similar, so their corresponding sides would be in proportion.
Therefore,
[tex] \frac{a}{b} = \frac{2.1}{1.4} \\ \\ \frac{a}{b} = \frac{3}{2} [/tex]
12. the number of errors in a textbook follows a poisson distribution with a mean of 0.04 errors per page. what is the expected number of errors in a textbook that has 204 pages? circle one answer.
The number of errors in a textbook follows a Poisson distribution with a mean of 0.04 errors per page. To find the expected number of errors in a textbook with 204 pages, we need to multiply the mean by the number of pages.
Expected number of errors = mean * number of pages = 0.04 * 204 = 8.16
Therefore, we can expect to find approximately 8 errors in a textbook that has 204 pages, based on the given Poisson distribution with a mean of 0.04 errors per page. It is important to note that this is only an expected value and the actual number of errors could vary.
Additionally, Poisson distribution assumes that the errors occur independently and at a constant rate, which may not always be the case in reality. Nonetheless, the Poisson distribution provides a useful approximation for the expected number of rare events occurring in a given interval.
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suppose we have 3 features in our task. if we apply polynomial regression with degree =3; how many features will be used in this model?
If we apply polynomial regression with degree =3 to a task with 3 features, a total of 20 features will be used in this model. This is because for each feature, we generate a polynomial combination with degree up to 3, resulting in a total of (3+3-1) choose 3 = 20 features.
If we apply polynomial regression with degree = 3 to a dataset with 3 features, then the resulting model will use a total of 20 features.
This is because polynomial regression with degree 3 involves creating new features by taking all possible combinations of the original features up to degree 3. In this case, we have 3 original features, so the number of new features created will be:
1 (constant term) + 3 (first-degree terms) + 32/2 (second-degree terms, since there are 3 features and we are taking combinations of 2) + 33*2/6 (third-degree terms, since there are 3 features and we are taking combinations of 3)
= 1 + 3 + 3 + 1 = 8 + 12 = 20
Therefore, the polynomial regression model with degree 3 applied to a dataset with 3 features will use 20 features in total.
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Un cliente tiene que devolver el dinero al banco donde solicitó un préstamo de $8 000.00 el 27 de diciembre. Si fue devuelto el 27 de octubre de ese mismo año y la tasa de descuento aplicada fue del 3% anual. ¿Cuál es el descuento que corresponde a esta operación? ¿Cuánto tiene qué pagar el cliente el 27 de octubre?
Answer:
694
Step-by-step explanation:
PLS HELP!! BRAINLIESt & 20 POINTS!! AND AN EXTRA THANKS AND 5 STARS
You survey your class to find out the types of pets they have at home.
Below are the results:
Dog, Dog, Cat, Cat, Cat, Dog, Hamster, Fish, Dog, Fish
a. Make a frequency table for the results. Continue on the following page.
b. Calculate the relative frequencies of the each animal type.
c. Using the relative frequencies explain which animals are most and least popular. Be specific and explain your reasoning.
Answer:
Step-by-step explanation:
a) Frequency table:
Category Frequency
Dog 4
Cat 3
Fish 2
Hamster 1
b) Relative frequencies of each animal type
Dog: 4/10 = 0.4
Cat: 3/10 = 0.3
Fish: 2/10 = 0.2
Hamster: 1/10 = 0.1
c) Popularity
Dog is the most popular because it has the highest relative frequency.
Hamster is the least popular because it has the lowest relative frequency.
The heights of a certain breed of dogs has a normal distribution with a mean of 28 inches and a standard deviation of 4 inches. If we randomly select 64 of these dogs, what is the probability that the mean height of 64 dogs is: a) Less than 27 inches? b) Greater than 28.5 inches? c) Between 27 and 28.5 inches?
The probability that the mean height of 64 dogs is between 27 and 28.5 inches is approximately 0.8531.
We can use the central limit theorem to approximate the distribution of the sample mean. The central limit theorem states that if we take a large enough sample from a population, the sample mean will be approximately normally distributed with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size. In this case, we have:
Population mean (μ) = 28 inches
Population standard deviation (σ) = 4 inches
Sample size (n) = 64
a) To find the probability that the mean height of 64 dogs is less than 27 inches, we need to standardize the sample mean and find the corresponding area under the standard normal distribution. We have:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
z = (27 - 28) / (4 / sqrt(64))
z = -2
Using a standard normal distribution table or calculator, we find that the probability of z being less than -2 is approximately 0.0228. Therefore, the probability that the mean height of 64 dogs is less than 27 inches is approximately 0.0228.
b) To find the probability that the mean height of 64 dogs is greater than 28.5 inches, we standardize the sample mean and find the area to the right of the standardized value. We have:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))
z = (28.5 - 28) / (4 / sqrt(64))
z = 1
Using a standard normal distribution table or calculator, we find that the probability of z being greater than 1 is approximately 0.1587. Therefore, the probability that the mean height of 64 dogs is greater than 28.5 inches is approximately 0.1587.
c) To find the probability that the mean height of 64 dogs is between 27 and 28.5 inches, we need to find the area under the standard normal distribution between the two standardized values. We have:
z1 = (27 - 28) / (4 / sqrt(64))
z1 = -2
z2 = (28.5 - 28) / (4 / sqrt(64))
z2 = 1
Using a standard normal distribution table or calculator, we find that the probability of z being between -2 and 1 is approximately 0.8531. Therefore, the probability that the mean height of 64 dogs is between 27 and 28.5 inches is approximately 0.8531.
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Decide whether statement (a) is true or false. Justify each answer. Assume A is an mn matrix and b is in mathbb R ^ m . a. The general least-squares problem is to find an x that makes Ax as close as possible to b. Choose the correct answer below. OA. The statement is false because the general least-squares problem attempts to find an x such that Ax = b O B. The statement is false because the general least-squares problem attempts to find an x that maximizes ||b - Ax|| . O C. The statement is true because the general least-squares problem attempts to find an x such that Ax = b OD. The statement is true because the general least-squares problem attempts to find an x that minimizes ||b - Ax||.
The statement is false because the general least-squares problem attempts to find an x that minimizes ||b - Ax||.
The general least-squares problem aims to find a solution for the equation Ax = b when there is no exact solution. In other words, it seeks to find an x that minimizes the residual vector ||b - Ax||.
The residual vector represents the error between the actual values of b and the values predicted by the matrix equation Ax. The objective is to minimize this error by finding the values of x that provide the best approximation to the equation.
The least-squares solution is obtained by minimizing the sum of the squared residuals, which is equivalent to minimizing the norm (magnitude) of the residual vector. Therefore, the goal is to find an x that minimizes the expression ||b - Ax||.
The statement (a) suggests that the general least-squares problem aims to find an x such that Ax = b, which is not correct. If Ax = b has an exact solution, then there is no need for the least-squares approach. The least-squares problem is specifically designed for cases where there is no exact solution.
Option A is incorrect because it contradicts the purpose of the least-squares problem. Option B is incorrect because it suggests maximizing the norm of the residual vector, which is not the objective. Option C is incorrect because it claims that the statement is true, but the statement is actually false. The correct answer is Option D, which correctly states that the general least-squares problem attempts to find an x that minimizes ||b - Ax||.
By minimizing the residual error, the least-squares solution provides the best approximation to the equation Ax = b in situations where an exact solution is not possible. This has important applications in various fields, including statistics, data fitting, signal processing, and regression analysis.
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Evaluate.
3^-3√8
a. -1/9
b. 91
c. -9
d. 1/9
The expression is evaluated to 1/9. Option D
How to determine the valueWe need to know that index forms are described as mathematical forms used in the representation of number or variables that are too large or too small in more convenient forms.
Also, other names for these index forms are scientific notation and standard forms.
From the information given, we have that;
[tex]3^-^\sqrt[3]{8}[/tex]
Now, find the cube root of the exponent with value of 8, we have;
∛8 = 2
Substitute the value, we have;
3⁻²
Express as a fraction, we get;
1/3²
Find the square of the denominator
1/9
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help meee plssssssss
Answer:
A) Triangle PQR
Step-by-step explanation:
This is the correct triangle because it has the exact same measurements as Triangle ABC. Angles A and P are the same angle on the triangles and they have the same measure (70). Angles C and R are the same angle on the triangles and they have the same measure (75). To solve for Angle C in Triangle ABC though, add angles 70 + 35 together to get 105. A triangle has a total measure of 180, so subtract 180 - 105 to get 75.
Hope it helps!
Duane And his family are playing a game Dwayne scores 11 points each tile matches 1/3 point model the solution with an equation how many tile matches this Dwayne make
Answer:
33 tiles
Step-by-step explanation:
Given
[tex]1\ tile = \frac{1}{3}\ points[/tex]
Required
Determine the number of tiles in 11 points
Represent this with x. So, we have:
[tex]1\ tile = \frac{1}{3}\ points[/tex]
[tex]x = 11\ points[/tex]
Cross Multiply
[tex]x * \frac{1}{3} = 11 * 1[/tex]
Multiply through by 3
[tex]3*x * \frac{1}{3} = 11 * 1*3[/tex]
[tex]x = 33[/tex]
A cookie jar starts off with 32 cookies in it, and
each day 2 cookies are eaten. After a certain
number of days, there are 14 cookies left in the jar.
Which of the following equations represents the
day, d, when there are 14 cookies in the jar?
Answer:
9 days
Step-by-step explanation:
32-2-2-2-2-2-2-2-2-2 = 14
The equation that represents the day, d, when there are 14 cookies in the jar will be 14 = 32 - 2d, and that day will be on the 9th day.
What is the equation?There are many different ways to define an equation. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of 2 mathematical expressions.
As per the given,
Initial cookie = 32
Since each day 2 cookies are eaten so after "d" days the total eaten cookies will be 2d.
Remaining cookies = initial cookies - eaten cookies
Remaining cookies = 32 - 2d
The day when 14 cookies remain will be,
14 = 32 - 2d
2d = 32 - 14
d = 9
Hence "The equation that represents the day, d, when there are 14 cookies in the jar will be 14 = 32 - 2d, and that day will be on the 9th day".
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2 Simplify -3(16 = 35+4)/5. Show your work.
Answer:
-9
Step-by-step explanation:
Parentheses first:16-35+4 = -35+16+4 = -35+20 = -15
3(-15) = -45/5
-45 divided by 5 = -9
A rectangle measures 2 2/3
inches by 2 1/3 inches. What is its area?
thank you!
Answer:
Step-by-step explanation:
To find the area of a rectangle, we multiply its length by its width.
The length of the rectangle is 2 2/3 inches, which can be expressed as an improper fraction: (3 * 2 + 2)/3 = 8/3 inches.
The width of the rectangle is 2 1/3 inches, which can also be expressed as an improper fraction: (3 * 2 + 1)/3 = 7/3 inches.
Now, we can calculate the area by multiplying the length and width:
Area = (8/3) * (7/3)
= (8 * 7)/(3 * 3)
= 56/9
Therefore, the area of the rectangle is 56/9 square inches, which can be simplified, if needed.
Consider the following integral. Sketch its region of integration in the xy-plane.
∫1_0∫y_(√y) 110x2y3dxdy
(a) Which graph shows the region of integration in the xy-plane? ? A B
(b) Evaluate the integral.
a. The region of integration is the region between the curves[tex]y = x^4[/tex]and [tex]y = x^2[/tex], for 0 ≤ y ≤ 1.
b. The value of the integral is -0.6875.
To sketch the region of integration, we need to find the boundaries of the integral.
We are integrating with respect to x first, so we need to express the limits of integration for x in terms of y.
From the inner integral, we have:
y ≤ [tex]x^2[/tex] ≤ √y
Taking the square root of both sides of the right inequality, we get:
[tex]y^{1/4}[/tex] ≤ x ≤ √y
Thus, the region of integration is the region between the curves[tex]y = x^4[/tex]and [tex]y = x^2[/tex], for 0 ≤ y ≤ 1.
To sketch the region, we can first draw the curves [tex]y = x^4[/tex] and [tex]y = x^2,[/tex]as shown in graph (A) below:
| .
1 | .
| .
| .
*------------>
0 1 2 3
The region of integration is the shaded region between the curves, as shown in graph (B) below:
| +
1 | +
| +++
| +++
*------------>
0 1 2 3
To evaluate the integral, we can use the boundaries we found and evaluate the integral using the following steps:
[tex]\int 1_0 \int y_(\sqrt{y } ) 110x^2y^3dxdy[/tex]
[tex]= \int 0^1 \int x^{1/2}^x^2 110x^2y^3 dy dx[/tex] (using the limits we found earlier)
[tex]= \int 0^1 110x^2 [(1/4)y^4]_(x^{1/2})^{x^2} dx[/tex] (integrating with respect to y)
[tex]= \int 0^1 110x^2 [(1/4)x^8 - (1/4)x^4] dx[/tex] (substituting the limits of integration)
[tex]= \int 0^1 (27.5x^10 - 27.5x^6) dx[/tex].
[tex]= [2.75x^11 - 3.4375x^7]_0^1[/tex](integrating and substituting limits)
= 2.75 - 3.4375.
= -0.6875.
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Samples of size 60 are drawn from a large population known to have a mean of 78 and a standard deviation of 12. What proportion of these samples would we expect to have a mean greater than 80?
a
0.566
b
0.098
c
Almost none
d
0.434
e
0.901
Answer: 0.098
Step-by-step explanation: First you’ll need to find the sample distribution mean and standard deviation (which is 78 and 1.55). Then you plug it into calculator using normal approximation cause n is large: normalcdf(80,100,78,1.55)
As, 0.098 proportion of these samples would, we expect to have a mean greater than 80. Thus, option (C) is correct.
What is the population?As the population is defined, as a group of groupings with a common characteristic. In data point, a population is the pool of individuals from which a statistical sample is drawn for a study. As we see, different countries have different populations.
As here the μ = 78, σ = 1.5492 and x = 80
We need to compute P(X >= 80). The corresponding z-value is calculated using Central Limit Theorem
z = (x - μ)/σ
z = (80 - 78)/1.5492 = 1.29
Therefore,
P(X >= 80) = P(z <= (80 - 78)/1.5492)
= P(z >= 1.29)
= 1 - 0.902 = 0.098
Therefore, The right option (B) is correct
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State if the triangles are similar. If so, state how you know they are.
Answer:
The triangles are similar
Step-by-step explanation:
Since EF ║ UT, m∠FUT = m∠EFU because alternate interior angles are congruent when lines are parallel.
m∠UDT = m∠EDF because vertical angles are congruetnt
Therefore ΔUDF is similar to ΔFDE because if two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.
Find m of arc JA
See photo below
The measure of the arc angle JA is 76 degrees.
How to find arc angle?The sum of angles in a cyclic quadrilateral is 360 degrees. The opposite angles in a cyclic quadrilateral is supplementary.
Therefore, Let's find the measure of arc angle JA.
26x + 1 = 1 / 2 (18x + 4 + 6 + 32x)
26x + 1 = 1 / 2 (50x + 10)
26x + 1 = 25x + 5
26x - 25x = 5 - 1
x = 4
Therefore,
arc angle JA = 18x + 4
arc angle JA = 18(4) + 4
arc angle JA =72 + 4
arc angle JA = 76 degrees.
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Please help due by tonight.
( . ) ( . )
\
\____/
Answer:
y = 1/2x^2 - 2x - 1
Step-by-step explanation:
Vertex(2, - 3)
Equation of Parabola (In vertex form): y = a(x - h)^2 + k,
y = a(x - 2)^2 - 3
Substitute given point (0, -1) and solve for a:
y = a(x - 2)^2 - 3,
-1 = a(0 - 2)^2 - 3,
a = 1/2
Equation: y = 1/2(x - 2)^2 - 3 = 1/2(x^2 - 4x + 4) - 3 = 1/2x^2 - 2x + 2 - 3 = 1/2x^2 - 2x - 1,
y = 1/2x^2 - 2x - 1: Option C
if the accaleration of an object is given by dv/dt=v/7, find the position function s(t) if v(0)=1 and s(0)= 2
Step-by-step explanation:
Integrate with respect to 't' the accel function to get the velocity function:
velocity = v/7 t + c1 when t = 0 this =1 so c1 = 1
velocity = v/7 t + 1 integrate again to find position function
s = v/14 t^2 + t + c2 when t = 0 this equals 2 so c2 = 2
s = v/14 t^2 + t + 2
( Let me know if this is incorrect and I will re-evaluate)
the sampling distribution of is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n1 and n2 are large. a. true b. false
The sampling distribution is normal if the sampled populations are normal, and approximately normal if the populations are nonnormal and the sample sizes n1 and n2 are large: (A) TRUE
The central limit theorem states that as sample sizes increase, the distribution of the sample means approaches a normal distribution regardless of the shape of the population distribution, as long as the samples are randomly selected and independent.
Therefore, if the populations from which the samples are drawn are normal, the sampling distribution of the means will also be normal.
However, even if the populations are nonnormal, the sampling distribution will still be approximately normal if the sample sizes are large enough.
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what is speaker? what is printer
Answer:
A speaker is a term used to describe the user who is giving vocal commands to a software program. ... A computer speaker is an output hardware device that connects to a computer to generate sound. The signal used to produce the sound that comes from a computer speaker is created by the computer's sound card
Step-by-step explanation:
rate nyoooo plssss
let A^2 = A. prove that either A is singular or det(A)=1
Eeither A is singular or det(A) = 1.
Let A be a square matrix such that A^2 = A.
If A is singular, then det(A) = 0, and we are done.
Otherwise, let B = A(I - A). Then we have:
B^2 = A(I - A)A(I - A) = A^2(I - A)^2 = A(I - A) = B
Multiplying both sides by B^-1 (which exists since B is invertible), we get:
B^-1 B^2 = B^-1 B
I = B^-1
Now we have:
det(A) = det(B)/det(I - A)
Since B = A(I - A), we have:
det(B) = det(A)det(I - A) = det(A)(1 - det(A))
Substituting into our expression for det(A), we get:
det(A) = det(A)(1 - det(A))/(1 - det(A))
Simplifying, we get:
1 = det(A)
Therefore, either A is singular or det(A) = 1.
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Natalie is a lab technician and earns $10.65 per hour. What are her wages for 40 hours?
pls help im in need
use the integral test to determine whether the series is convergent or divergent. [infinity] ∑ 14/n^10 n = 1
The integral ∫1 to infinity 14/x¹⁰ dx converges, the series ∑ 14/n¹⁰ converges by the integral test.
The integral test to determine whether the series is convergent or divergent.
The integral test states that if f(n) is a continuous, positive, and decreasing function on [1, infinity), and if the series ∑ f(n) is convergent, then the series ∑ a(n) is also convergent, where a(n) = f(n) for all n.
Let f(n) = 14/n¹⁰.
Then f(n) is continuous, positive, and decreasing on [1, infinity).
To apply the integral test, we need to evaluate the integral
∫1 to infinity 14/x¹⁰ dx.
Using the power rule of integration, we have
∫1 to infinity 14/x¹⁰ dx = [(-14/9)x⁻⁹] from 1 to infinity
= [-14/(9 ×(infinity)⁹)] - (-14/9)
= 14/9.
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The given series, Σ(14/n^10) from n = 1 to infinity, is convergent.
To determine the convergence of the series using the integral test, we compare it to the integral of the corresponding function. Let's integrate the function f(x) = 14/x^10:
∫(14/x^10) dx = -14/(9x^9)
Now, we evaluate the definite integral from 1 to infinity:
∫[1,∞] (14/x^10) dx = lim[a→∞] (-14/(9x^9)) - (-14/(9(1^9)))
= 14/9
The integral of the function converges to a finite value of 14/9. According to the integral test, if the integral of the corresponding function is convergent, then the series is also convergent. Therefore, the series Σ(14/n^10) from n = 1 to infinity is convergent. In conclusion, the given series is convergent.
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