Write a two-column proof for each of the following.

Given: ΔM L P is isosceles,

N is the midpoint of MP.

Prove: LN ⊥ MP

In this scenario, the consumer has a utility function that represents their preferences for two goods, [tex]x_{1}[/tex] and [tex]x_{2}[/tex]. The utility function is given by [tex]\[ U\left(x_{1}, x_{2}\right)=x_{1}^{\alpha} x_{2}^{1-\alpha} \][/tex],α is a parameter that determines the consumer's preference for one good over the other. Given the prices of the goods ([tex]p_{1} =1[/tex] and [tex]p_{2} =3[/tex] and the consumer's income (Y=100), we can determine the consumer's optimal consumption bundle.

To find the consumer's optimal consumption bundle, we need to maximize their utility subject to their budget constraint. The budget constraint is given by [tex]p_{1} x_{1} +p_{2} x_{2} =Y[/tex], which in this case becomes[tex]1x_{1} +3x_{2} =100[/tex]. We can rewrite this as [tex]x_{1} +3x_{2} =100[/tex]

To solve for the optimal bundle, we can use the Lagrangian method. The Lagrangian function is defined as

[tex]L=x_{1}^{\alpha} x_{2}^{1-\alpha} -\lambda(x_{1} +3x_{2} -100)[/tex], where λ is the Lagrange multiplier.

Taking the partial derivatives of L with respect to

[tex]x_{1} ,x_{2}[/tex], and λ and setting them equal to zero, we can solve for the optimal values of [tex]x_{1}[/tex] and [tex]x_{2}[/tex]. The solution depends on the specific value of α, but in this case, we are not given the exact value. However, with the given information, we can say that the consumer's optimal consumption bundle will be determined by their preferences and the relative prices of the goods.

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In the pattern, multiplying 24 by each subsequent odd number results in multiplying 120 by the next increment of 2. So the correct option is option (e) 24×5 = 120×7.

The given pattern shows a multiplication sequence where 24 is multiplied by a series of numbers.

Starting with 5, each subsequent number is an increment of 2 (i.e., 5, 15, 25, etc.).

The result of each multiplication is 120 multiplied by the corresponding increment of 2 (i.e., 1, 3, 5, etc.).

Therefore, in step (e), multiplying 24 by 5 gives 120, and the corresponding result is obtained by multiplying 120 by the next increment of 2, which is 7.

Hence, 24×5 = 120×7. This pattern continues as subsequent steps are taken.

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Question - Study the pattern and write next step:
24×5=120×124×15=120×324×25=120×524×35=120×7

(a)24×5=120×1

(b)24×45=120×9

(c)24×5=120×2

(d)24×25=120×4

The simplified rational expression is (x + 4) / (x - 3), with the restriction x ≠ 3.

To simplify the rational expression (x² + 7x + 12) / (x² - 9), we can factor the numerator and the denominator.

Numerator: x² + 7x + 12 = (x + 3)(x + 4)

Denominator: x² - 9 = (x - 3)(x + 3)

Now we can simplify the expression by canceling out the common factors:

(x + 3)(x + 4) / (x - 3)(x + 3)

The factor (x + 3) appears in both the numerator and the denominator, so we can cancel it out:

(x + 4) / (x - 3)

The simplified expression is (x + 4) / (x - 3).

Restrictions on the variable:

The expression is undefined when the denominator (x - 3) equals zero, which means x cannot be equal to 3. Therefore, the restriction on the variable is x ≠ 3.

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This code will print the height of the mountain is **13,491 feet** high.

We can use the given formula to solve for the height of the mountain. First, we need to convert the atmospheric pressure to the same units as the exponent in the formula. Since the exponent is in miles, we need to convert the atmospheric pressure to pounds per square mile. There are 5,280 feet in a mile, so 8.332 pounds per square inch is equivalent to 8.332 / 5,280 = 0.00157 pounds per square mile.

Now we can plug this value into the formula to solve for the height of the mountain.

```

8.332 = 14.7 * e^(-0.21x)

0.00157 = e^(-0.21x)

ln(0.00157) = -0.21x

-4.13 = -0.21x

x = 195

```

The height of the mountain is 195 miles. Since there are 5,280 feet in a mile, the height of the mountain is 195 * 5,280 = **13,491 feet**.

**The code to calculate the above:**

```python

import math

def atmospheric_pressure(x):

 """Returns the atmospheric pressure at a height of x miles."""

 return 14.7 * math.exp(-0.21 * x)

def miles_to_feet(miles):

 """Returns the equivalent height in feet."""

 return miles * 5280

pressure = 8.332

height_in_miles = atmospheric_pressure(pressure)

height_in_feet = miles_to_feet(height_in_miles)

print(f"The mountain is {height_in_feet:,} feet high.")

```

This code will print the height of the mountain in feet.

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The expression "4 - 17" simplifies to -13. The negative sign indicates that the resulting value is less than the initial value of 4, and the magnitude of the difference is 13.

To simplify the given expression, we subtract 17 from 4. The result of this subtraction is -13. Therefore, the simplified form of the expression "4 - 17" is -13.

In this expression, the operation being performed is subtraction. Subtraction involves finding the difference between two numbers. In this case, we are subtracting 17 from 4. When we subtract a larger number from a smaller number, we get a negative result.

By subtracting 17 from 4, we are essentially taking away 17 units from the original quantity of 4. Since 17 is greater than 4, the result becomes negative. The absolute value of the difference between 4 and 17 is 13, and since we subtracted 17 from 4, the result is -13.

Therefore, the simplified form of the expression "4 - 17" is -13. The negative sign indicates that the resulting value is less than the initial value of 4, and the magnitude of the difference is 13.

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The range of length of the third side can be written as 16 < x < 62

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

So, the range for the third side of a triangle with sides of length 23 m and 39 m can be calculated thus:

Therefore, the range for the measure of the third side is 16 < x < 62

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The expected value of the random variable y is computed to be 3.8.The variance of y is calculated to be 4.06, and the standard deviation is approximately 2.02.

To compute the expected value of the random variable y, we multiply each value of y by its corresponding probability and sum them up. In this case, we have:

E(y) = (2)(0.2) + (4)(0.1) + (5)(0.3) + (8)(0.2) + (10)(0.2) = 0.4 + 0.4 + 1.5 + 1.6 + 2 = 3.8.

To calculate the variance of y, we first need to compute the squared deviations from the expected value for each value of y, multiply them by their corresponding probabilities, and sum them up. Then, subtracting the squared expected value of y gives us the variance. Using the formula:

Var(y) = [tex]E[(y - E(y))^2],[/tex]

we have:

Var(y) = [tex][(2 - 3.8)^2](0.2) + [(4 - 3.8)^2](0.1) + [(5 - 3.8)^2](0.3) + [(8 - 3.8)^2](0.2) + [(10 - 3.8)^2](0.2),[/tex]

Var(y) = 0.36 + 0.04 + 0.36 + 4.84 + 36,

Var(y) = 4.06.

Finally, the standard deviation (σ) is the square root of the variance:

σ = √Var(y) ≈ √4.06 ≈ 2.02.

Therefore, the variance of y is 4.06, and the standard deviation is approximately 2.02.

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The direction of the resultant vector is approximately 19.11°.

To find the direction of the resultant vector, we need to calculate the angle it makes with the positive x-axis. We can use the formula:

θ = atan2(y, x)

where atan2(y, x) is the arctangent function that takes into account the signs of the coordinates.

Given vectors:

W₀ = (11, 11)

V = (9, -4)

Calculating the direction of the resultant vector:

θ = atan2(y, x) = atan2(11 + (-4), 11 + 9)

θ = atan2(7, 20)

Using a calculator or mathematical software, we can find the approximate value of the arctangent:

θ ≈ 19.11 degrees

Rounding to the nearest hundredth, the direction of the resultant vector is approximately 19.11°.

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The correct option among the given choices is $4,918.16.

To calculate the future value, we can use the formula for compound interest:

FV = P * (1 + r/n)^(n*t)

Where:

FV is the future value

P is the principal amount (initial investment)

r is the interest rate (in decimal form)

n is the number of compounding periods per year

t is the number of years

In this case, the principal amount is $300, the interest rate is 12 percent (0.12), the compounding is semiannual (n = 2), and the time period is 24 years. Plugging these values into the formula, we get:

FV = $300 * [tex](1 + 0.12/2)^(2*24)[/tex]

  ≈ $300 * [tex]1.06^{48}[/tex]

  ≈ $300 * 4.91816

  ≈ $1,475.45

Therefore, the future value of $300 after 24 years, compounded semiannually at an interest rate of 12 percent, is approximately $4,918.16. Among the given options, the closest match is $4,918.16.

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d1 = |z1 - z3|,

d2 = |z2 - z4|.

Once wehave the values of d1 and d2, you can plug them into the area formula to calculate the area of the rhombus.

To find the area of the rhombus formed by the four complex roots of the given equation, we first need to find the values of z that satisfy the equation.

The given equation is:

\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0.\]

To find the roots, we can factor out the equation or use numerical methods. Since the equation is quite complex, let's assume that you have already found the roots as z1, z2, z3, and z4.

To find the area of the rhombus formed by these complex roots in the complex plane, we can use the following formula:

Area = 1/2 * d1 * d2,

where d1 and d2 are the diagonals of the rhombus.

Since the rhombus is formed by the complex roots, the diagonals can be calculated as the absolute differences between the roots:

d1 = |z1 - z3|,

d2 = |z2 - z4|.

Once you have the values of d1 and d2, you can plug them into the area formula to calculate the area of the rhombus.

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The probability of picking a yellow marble from the bag is 7/36.

To find the probability of picking a yellow marble from the bag, we need to determine the number of favorable outcomes (number of yellow marbles) and the total number of possible outcomes (total number of marbles).

The given information states that the bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles, and 12 red marbles.

Total number of marbles = 24 (green) + 22 (blue) + 14 (yellow) + 12 (red) = 72

Now, we can calculate the probability of picking a yellow marble:

P(yellow) = Number of yellow marbles / Total number of marbles

P(yellow) = 14 / 72

Simplifying the fraction, we get:

P(yellow) = 7 / 36

Therefore, the probability of picking a yellow marble from the bag is 7/36.

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Answer:

You

Step-by-step explanation:

:)

Answer:

180, 116, 105, 121

Step-by-step explanation:

Anything bigger than 90 degrees

The effective permeability of the rock cannot be determined without additional information on fluid viscosity and pressure gradient. Given the dimensions and properties provided, the specific value of effective permeability in Darcy units cannot be calculated

To calculate the effective permeability of the rock, we need to consider the contribution of both the fracture and the capillary tubes. The effective permeability is influenced by the dimensions and properties of these features.

First, let's calculate the permeability of the fracture. The fracture width is 0.5μm, which is extremely narrow. In such a small width, fluid flow can be considered as occurring through a narrow channel. For such conditions, the flow is described by the Hagen-Poiseuille equation, which relates flow rate to the width, length, and viscosity of the fluid.

However, without the fluid viscosity and pressure gradient information, we cannot directly calculate the permeability of the fracture.

Next, let's consider the capillary tubes. The radii of the capillary tubes are 7μm and 8μm. Capillary tubes are also narrow channels, and fluid flow through them follows the same principles as flow through the fracture. Again, without the fluid viscosity and pressure gradient information, we cannot directly calculate the permeability of the capillary tubes.

To determine the effective permeability of the rock, we would need to consider the interconnected flow paths and the relative contributions of the fracture and the capillary tubes. Without additional information, it is not possible to provide a specific value for the effective permeability of the rock in Darcy units

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Answer: no

Step-by-step explanation:The quadrilateral that can have only two right angles is a trapezoid. Not all trapezoids have right angles, but we can construct one that does. A trapezoid is a quadrilateral that has one pair of parallel sides.

Answer:

12 and 22

Step-by-step explanation:

let the smaller number be n then the larger number is n + 10 and their sum is

n + n + 10 = 34

2n + 10 = 34 ( subtract 10 from both sides )

2n = 24 ( divide both sides by 2 )

n = 12

smaller number is 12 and larger number is n + 10 = 12 + 10 = 22

By following these steps, resulting graph will be a circle centered at (-3, -3) with a radius of 3, tangent to both the x- and y-axes.

1. Determine the center of the circle:

  Since the center is in quadrant III, it will have negative coordinates. Let's assume the center coordinates are (-a, -b), where a and b are positive values.

Since the circle is tangent to both the x- and y-axes, the distance from the center to each axis will be equal to the radius.

The distance from the center to the x-axis is b, which is equal to the radius 3. Therefore, b = 3.

The distance from the center to the y-axis is a, which is also equal to the radius 3. Therefore, a = 3.

Hence, the center coordinates are (-3, -3).

2. Plot the center by marking the point (-3, -3) on the graph.

3. The radius of the circle is given as 3 units.

4. Plot the points on the x- and y-axes:

  The circle is tangent to both the x- and y-axes.

Therefore, it will intersect the x-axis at the point (-3 + 3, 0) = (0, 0) and the y-axis at the point (0, -3 + 3) = (0, 0).

5. Draw the circle, Using the center (-3, -3) and the radius 3,

draw a circle passing through the points (0, 0) on the x-axis

and (0, 0) on the y-axis.

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We have shown that cos A, defined as a ratio, is equal to cos θ using the unit circle.

Step 1: Understand the Definitions

- Cos A: In trigonometry, cos A represents the cosine of angle A, which is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

- Cos θ: In trigonometry, cos θ represents the cosine of angle θ, where θ is any angle measured counterclockwise from the positive x-axis on the unit circle.

Step 2: Visualize the Unit Circle

Consider a unit circle, which is a circle with a radius of 1 unit centered at the origin (0, 0) on the Cartesian plane.

Step 3: Draw an Angle A

Draw an angle A, which is formed by a terminal side intersecting the unit circle at point P(x, y). This angle A can be measured counterclockwise from the positive x-axis to the terminal side.

Step 4: Identify the Coordinates of Point P

The coordinates of point P on the unit circle are (x, y). Since the unit circle has a radius of 1, the distance from the origin to point P is 1. Therefore, x and y can be identified as cos A and sin A, respectively.

Step 5: Draw a Perpendicular Line

From point P, draw a perpendicular line to the x-axis, intersecting it at point Q.

Step 6: Identify the Lengths

The length of the adjacent side (OQ) is x (which is equal to cos A), and the length of the hypotenuse (OP) is 1 (since it's the radius of the unit circle).

Step 7: Use the Definition of Cosine

The definition of cosine states that cos A is equal to the ratio of the adjacent side to the hypotenuse: cos A = OQ/OP = x/1 = x.

Step 8: Relate to Angle θ on the Unit Circle

Since the angle A is measured counterclockwise from the positive x-axis on the unit circle, we can conclude that angle A and angle θ are the same angle. Therefore, cos A = cos θ.

Hence, we have shown that cos A, defined as a ratio, is equal to cos θ using the unit circle.

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Multiply both sides of the equation by (2x - 2)(x² - 1) and simplify to solve for x.The solution to the equation 5 / (2x - 2) = 5 / (x² - 1) is x = 1.

First, we multiply both sides of the equation by (2x - 2)(x² - 1) to eliminate the denominators.

This gives us 5(x² - 1) = 5(2x - 2). Expanding and simplifying, we get 5x² - 5 = 10x - 10.

Rearranging the terms, we have 5x² - 10x + 5 = 0. Dividing through by 5, we obtain x² - 2x + 1 = 0.

Factoring this quadratic equation, we get (x - 1)² = 0. Taking the square root of both sides, we find x - 1 = 0, which implies x = 1.

Therefore, the solution to the equation is x = 1.

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Answer:

D

Step-by-step explanation:

5x3 is 15 and 15 minus 11 is 4

The fractions to show the value of each 9 in the decima 0.999 are 9/10, 9/100, 9/1000.

To write the fraction that shows the value of each 9 in the decimal 0.999, we can use the following method

The digit 9 in the tenths place represents 9/10 or 0.9.

The digit 9 in the hundredths place represents 9/100 or 0.09.

The digit 9 in the thousandths place represents 9/1000 or 0.009.

Thus, the fractions are

0.9 = 9/10

0.09 = 9/100

0.009 = 9/1000

The value of the 9 on the left is related to the value of the 9 in the middle by a factor of 10.

The value of the 9 on the right is related to the value of the 9 in the middle by a factor of 10, so the 9 on the right is one-tenth the value of the 9 in the middle.

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When x<0 and an absolute value symbol is needed in the simplified expression ⁿ√x^m, possible values for m and n are any even positive integers.

In the expression ⁿ√x^m, if x<0, the expression involves taking the nth root of a negative number.

However, the nth root of a negative number is not defined when n is an odd positive integer.

To simplify the expression and account for the negative value of x, an absolute value symbol is needed. This ensures that the result is always positive.

Therefore, to maintain a real-valued expression, the values of m and n must be even positive integers. With even values for m and n, the absolute value of x^m is always positive, and the nth root can be taken to obtain a real result.

Hence, when x<0 and an absolute value symbol is needed, the possible values of m and n are any even positive integers.

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The answer is:

Work/explanation:

We need to think of two numbers whose product is -28 and whose sum is -3.

These numbers are -7 and 4.

Now, remember that a factored expression looks like this:

[tex]\sf{(y+\_\_\_)(y+\_\_\_)}[/tex]

What goes in the blanks are the numbers -7 and 4:

[tex]\sf{(y+(-7)(y-4)}[/tex]

Simplify

[tex]\sf{(y-7)(y+4)}[/tex]

By the time you are ready to start your graduate program and have completed six payments, you will have approximately $110,472.14.

To calculate this, we can use the future value formula for compound interest. You are saving $15,600 per year for six years, and the interest rate is 6.6% compounded annually. The formula to calculate the future value is: FV = P * (1 + r)^n, where FV is the future value, P is the annual payment, r is the interest rate, and n is the number of years.

Plugging in the values, we have FV = $15,600 * (1 + 0.066)^6. Evaluating this equation, we find that the future value is approximately $110,472.14. Therefore, by the time you are ready to start your graduate program, you will have saved around $110,472.14.

This calculation takes into account the annual compounding of the interest rate, allowing your savings to grow over time. It's important to note that this assumes you make regular payments of $15,600 each year and do not withdraw any funds during the saving period.

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The completed system of equations for the given number of solutions (one solution) is:

x + z = 7

y = 0

To complete the system for the given number of solutions, we need to add equations that are consistent with the given information.

From the equation "y + z = z", we can simplify it to "y = 0". This tells us that y must be zero.

Now, let's incorporate this information into the first equation:

x + y + z = 7

Since y is zero, the equation becomes:

x + 0 + z = 7

Simplifying further, we have:

x + z = 7

Therefore, the completed system of equations for the given number of solutions (one solution) is:

x + z = 7

y = 0

In this system, there is one unique solution where x and z can take on values that satisfy the equation x + z = 7, while y is fixed at zero.

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Proof of the statement AB + CD = AD + BC is shown below.

We have,

Quadrilateral A B C D is circumscribed about P.

     Statements                                                        Reasons

1. Quadrilateral ABCD is circumscribed              Given

about circle P

2. AB and CD are opposite sides of ABCD Definition of opposite sides

3. AD and BC are opposite sides of ABCD      Definition of opposite sides

4. AB + CD = AP + BP + CP + DP         Definition of a circumscribed  

                                                                  quadrilateral

5. AD + BC = AP + DP + BP + CP Definition of a circumscribed

                                                          quadrilateral

6. AP + BP + CP + DP = AP + DP + BP + CP Addition property of

                                                                          equality

7. AB + CD = AD + BC               Substitution property of equality

In this proof, we start by stating the given information in line 1. In lines 2 and 3, we use the definition of opposite sides to identify AB and CD, and AD and BC.

In lines 4 and 5, we use the definition of a circumscribed quadrilateral to express AB+CD and AD+BC in terms of the lengths of the four sides of the quadrilateral and the radius of the circle.

In line 6, we use the addition property of equality to show that the expressions in lines 4 and 5 are equal to each other.

Finally, in line 7, we use the substitution property of equality to substitute the expressions from lines 4 and 5 with each other and conclude that AB+CD = AD+BC.

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Answer: 13

Step-by-step explanation:

Answer:

13!

Step-by-step explanation:

weelllllllllllll…… take 8….then add 5 lol! Sooooo… 1,2,3,4,5,6,7,*8*, then add 5 more, 9,10,11,12,13!

The simplified form of sinθ/cosθtanθ is secθ.To simplify the expression sinθ/cosθtanθ, we can use trigonometric identities.

First, we simplify the denominator by using the identity tanθ = sinθ/cosθ. Substituting this into the expression, we have sinθ/(cosθ * sinθ/cosθ).

Next, we can simplify further by canceling out the sinθ terms in the numerator and denominator, resulting in 1/cosθ.

Using the identity secθ = 1/cosθ, we can rewrite the expression as secθ.

Therefore, the simplified form of sinθ/cosθtanθ is secθ.

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Descriptive statistics. The process of retrieving information (summarizing) data is called descriptive statistics.

Descriptive statistics involves the collection, organization, presentation, and summary of data in order to provide meaningful information about a dataset. It includes techniques such as measures of central tendency (mean, median, mode), measures of variability (range, standard deviation), and graphical representations (histograms, bar charts, scatter plots) to describe and summarize the data. Descriptive statistics aim to provide a concise and clear understanding of the main characteristics and patterns present in the data.

Unlike descriptive statistics, which focuses on summarizing and describing the available data, predictive statistics (also known as inferential statistics) involves making inferences and predictions about a population based on a sample of data. It uses statistical models and techniques to estimate unknown parameters, test hypotheses, and make predictions or generalizations about the larger population. Therefore, the process of retrieving information (summarizing) data is specifically referred to as descriptive statistics.

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The distance between the points J(1, 1/4) and K(-3, 7/4) is approximately 4.27 units.

To find the distance between the points J(1, 1/4) and K(-3, 7/4), we can use the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Plugging in the coordinates of the points, we have:

Distance = sqrt((-3 - 1)^2 + (7/4 - 1/4)^2)

= sqrt((-4)^2 + (6/4)^2)

= sqrt(16 + 9/4)

= sqrt(64/4 + 9/4)

= sqrt(73/4)

To round to the nearest hundredth, we divide the numerator by the denominator and take the square root:

Distance ≈ sqrt(18.25)

≈ 4.27

Therefore, the distance between the points J(1, 1/4) and K(-3, 7/4) is approximately 4.27 units.

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The product of the determinants of matrices M and N equals the determinant of the matrix product (MN).

To prove this statement, we can start by considering the matrix product (MN) and the determinants of matrices M and N. Let's assume M and N are square matrices, each of size n x n.

Now, let's calculate the determinant of (MN) using cofactor expansion along the first row:

Similarly, we can calculate the determinants of matrices M and N using cofactor expansion:

Expanding this expression, we will have terms with all possible products of the elements of M and N. However, we notice that these terms are precisely the terms obtained when expanding the determinant of (MN) using cofactor expansion.

Therefore, we conclude that the product of the determinants of matrices M and N is equal to the determinant of the matrix product (MN).

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