The product of two n×n orthogonal matrices, PQ, is also an orthogonal matrix.
To show that PQ is an orthogonal matrix, we need to demonstrate two properties: it is a square matrix and its transpose is equal to its inverse.
Square Matrix: Since P and Q are n×n orthogonal matrices, their product PQ will also be an n×n matrix, satisfying the condition of being square.
Transpose and Inverse: We know that P and Q are orthogonal matrices, so P^T = P^(-1) and Q^T = Q^(-1). Taking the transpose of PQ, we have (PQ)^T = Q^T P^T.
To show that (PQ)^T = (PQ)^(-1), we need to prove that (Q^T P^T)(PQ) = I, where I represents the identity matrix.
(Q^T P^T)(PQ) = Q^T (P^T P) Q
Since P and Q are orthogonal matrices, P^T P = I and Q^T Q = I.
Substituting these values, we have:
(Q^T P^T)(PQ) = Q^T I Q = Q^T Q = I
Therefore, (PQ)^T = (PQ)^(-1), showing that PQ is an orthogonal matrix.
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What is midpoint formula Class 11?
The midpoint formula is a mathematical formula used to find the midpoint of a line segment, given the coordinates of the two endpoints of the segment. In class 11, it is defined as:
Midpoint = ( (x1 + x2)/2 , (y1 + y2)/2 )
Where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment, and (x, y) are the coordinates of the midpoint of the segment.
For example, if the two endpoints of a line segment are (5,6) and (9,11)
Midpoint = ( (5 + 9)/2 , (6 + 11)/2 )
Midpoint = (7,8.5)
The midpoint of the line segment with endpoints (5,6) and (9,11) is (7,8.5)
It's worth noting that this formula can be adapted for three-dimensional points, where the midpoint formula would be
((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
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I'm actually really stuck will anyone help
Answer:
Step-by-step explanation:
104 units
The scatter plot shows a hiker's elevation above sea level during a hike from the base to the
top of a mountain. The equation of a trend line for the hiker's elevation is y=9.16x +659, where x
represents the number of minutes and y represents the hiker's elevation in feet. Use the equation of
the trend line to estimate the hiker's elevation after 150 minutes.
The trend line equation to estimate the hiker's elevation after 150 minutes is 2033 feet.
What is the hiker's elevation?Given that,
A trend line's equation for the hiker's elevation is y = 9.16x + 659.
We must determine,
The trend line equation is used to calculate the hiker's elevation after 150 minutes.
In response to the question,
A trend line's equation for the hiker's elevation is y = 9.16x + 659.
Where x is the number of minutes and y is the elevation of the hiker in feet.
Substitute the value of x = 150minute in the given equation to find the trend line to estimate the hiker's elevation after 150minutes.
Therefore,
y = 9.16x + 659
x = 150
y = 9.16(150) + 659
y = 1374 + 659
y = 2033
As a result, the trend line equation for estimating the hiker's elevation after 150 minutes is 2033 feet.
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The perimeter (y) of a given shape is a function of its longest side (x). The equation for this function is y = 2.5x + 14. What does a solution of (15 feet, 51.5 feet) represent in the context of the problem?
The solution of (15 feet, 51.5 feet) represents the perimeter of a shape with a longest side of 15 feet.
What is Equation?Equation is a mathematical statement that express the equality of two expression use in equal sign.
The equation for this perimeter can be written as y = 2.5x + 14, where y is the perimeter of the shape and x is the longest side of the shape.
Thus, when x is 15 feet, the perimeter of the shape is 51.5 feet, as represented by the solution (15 feet, 51.5 feet).
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What is the formula of triangle and square?
The formula for a triangle is the area of a triangle = ½ × base × height. The formula for a square is the area of a square = side2.
This formula is used to calculate the area of any triangle when the base and height are known. For example, if the base of a triangle is 10 cm and the height is 5 cm, the area of the triangle is 25 cm2.
The formula for a square is the area of a square = side2. This formula is used to calculate the area of any square when the side length is known. For example, if the side of a square is 10 cm, the area of the square is 100 cm2. To calculate this, we can use the formula by squaring 10 (102 = 100).
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2y - 2 = 4x
y = -x + 4
i need the solution please help only have a day to answer it
Answer: x=1,y=3
Step-by-step explanation:
Given,
y - 1 =2x
y = 2x + 1-----(i)
Again,
We have,
y = 2x + 1
Now replacing the value of y from (i)
2x + 1= -x +
or, 2x + x= 3
or,3x= 3
So x=1
Now,
y=2x +1
or, y=2×1+1
so y=
Answer:
x=1, y=3. (1, 3).
Step-by-step explanation:
2y-2=4x
y=-x+4
----------
2y-2=4x implies 2(y-1)=2(2x), so y-1=2x.
(-x+4)-1=2x
-x+4-1=2x
-x+3=2x
3=2x-(-x)
3=2x+x
3=3x
x=3/3
x=1
y=-1+4=3
THE FIRST PERSON TO ANSWER THIS QUESTION CORRECTLY NO NEED TO EXPLAIN GETS BRAINIEST!!(REALLY EASY)
Jenna has 34 pennies. Mandy has 48 pennies. Mandy says that 72 pennies in all. what should Mandy do to fix her mistake
2 3/4 is between the whole number_____and_____
Answer: 2 and 3
Step-by-step explanation:
Help Answer this math question
Answer:
c.
Step-by-step explanation:
Evaluate the following sin6A-cos6A
On solving the provided question we can say that the value of trigonometry value is - [tex](sin^2x - cos^2x )( 1- sin^2xcos^x)\\[/tex]
what is trigonometry?The area of mathematics known as trigonometry examines the correlation between triangle side lengths and angles. The area first appeared in the Hellenistic era, around the third century BC. from the use of geometry in astronomical study. The area of mathematics known as exact methods deals with specific trigonometric functions and how they might be used in calculations. There are six popular trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their respective names and acronyms (csc). Studying the characteristics of triangles, particularly right triangles, is called trigonometry. The study of geometry, however, is the characteristics of all geometric figures.
[tex]sin^6 x - cos^6x\\(sin^3x)^2 - (cos^3x)^2\\(sin x - cosx)(sinx + cosx)(1 + sinx cosx)(1 - sinx cosx)\\(sin^2x - cos^2x )( 1- sin^2xcos^x)\\[/tex]
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Part A Without computing, the sum of 2, 3, and -2 is , 1 of 1.. Part B Choose the explanation that supports your answer in Part A. Multiple choice question. cross out A) 2 and -2 are additive inverses and the sum of any number and its additive inverse is zero. The integer 3 is positive, so the sum with be positive. cross out B) There are two positive integers and one negative integer. There are more positive integers than negative integers, so he sum will be positive. cross out C) 2 and 3 multiplied together is 6. Then, 6 and -2 multiplied together is -12. Therefore, the sum of the three integers is negative. cross out D) If there are an odd number of negative integers then the sum is negative. One of the integers is negative, so the sum is negative.
A. Without computing, the sum of 2, 3, and -2 is 3.
B. The explanation that supports the answer in Part A is (A) 2 and -2 are additive inverses and the sum of any number and its additive inverse is zero. The integer 3 is positive, so the sum with be positive.
What is additive inverses?An additive inverse of a number is the value that results in zero when added to the original number. It is the value that we add to a number in order to get zero. If an is the original number, then its additive inverse is minus a, i.e., -a, such that;
a+(-a) = a – a = 0
By changing the sign of a number, we can find its additive inverse. A positive number's additive inverse is negative, while a negative number's additive inverse is positive. Except for the sign, the numerical value will not change. The additive inverse of 8 is -8, for example, while the additive inverse of -6 is 6.
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Which type of dilation are the given scale factors?
Select Expansion or Contraction to correctly describe the type of dilation for each given scale factor.
Scale Factor Expansion
336
554
0.5
-1.2
O
Contraction
O
O
O
O
ہے
Enlarging or contracting a figure's size is known as dilatation. In other words, a scale factor is used to either enlarge or diminish the image while maintaining similarity between the preimage and the image.
What Is A Dilation ?In line with the illustrations below. A dilation that produces a smaller image is called a reduction (conjure up shrinking), whereas a dilation that produces a larger image is called an enlargement (conjure up stretching).Image reduction occurs when the scale factor is between 0 and 1.An expansion of the image occurs when the scale factor exceeds 1.The center point of the dilation is located, and the distances between this center point and various points on the preimage and the image are measured in order to determine the scale factor for the dilation. According to Math Bits Notebook, we can determine the scale factor by dividing these distances by two.Given that each angle is congruent, each segment is proportional, the slope of each segment is preserved, and the perimeter of the preimage and image have the same scale factor, using dilation scale factors allows us to enlarge or reduce a figure to the desired size.To Learn more About dilatation Refer To:
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A survey asks 50 people if they prefer to stream movies or rent dvds. The result of the survey indicate that 25 people prefer to stream, 20 people prefer to rent dvds and 5 people had no preference what is the ratio of people who prefer streaming to the people surveyed
The ratio of people who prefer streaming to the people surveyed is 25:50, or 1:2.
What is ratio?Ratio is a comparison of two or more numbers or quantities that indicates the relative size of the quantities. Ratios are expressed in the form of a fraction or decimal, and they are used in mathematics, statistics, and other sciences. Ratios can be used to compare the sizes of different objects or values, or to compare the relationships between different numbers. Ratios are often used to measure the efficiency of a process or the relationship between different variables.
This ratio indicates that half of the people surveyed prefer to stream movies, while the other half prefer to rent DVDs.
Of the 50 people surveyed, 5 had no preference, meaning that 45 out of the 50 people had a preference for either streaming or renting DVDs.
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Use the diagram to determine which triangle similarity theorem is the most appropriate to justify the similarity relationship.
ABC DEF by ___________ Similarity.
Both triangles ABC and DEF are similar to each other based on angle angle similarity
What are similar triangles?Two triangles are said to be similar if the ratio of the corresponding sides of both triangles are proportional to each other. Also, all the three angles of one triangle are equal to that of the other triangle.
The angle angle (AA) similarity theorem states that two triangles are similar if two angles of one triangle is equal to that of the other triangle.
Both triangles ABC and DEF are similar to each other
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Use the Triangle Sum Theorem to find the measure of each angle. Drag and drop the angle measures into the correct boxes to show the measure of each angle.
Answer:
Step-by-step explanation:
so basically Use the Triangle Sum Theorem to find the measure of each angle. Drag and drop the angle measures into the correct boxes to show the measure of each angle.
your welcome lol. -_-
100 points + Brainliest Let C represents the cost in dollars and R represents the revenue in dollars. What is the break-even point? Use a table to help if necessary.
C=36x+200
R=76x
Break-even point: ([],[])
Answer:
The break even point is 380.
Step-by-step explanation:
36x + 200 = 76x
200 = 76x - 36x
200 = 40x
200/40 = x
5 = x
C = 36(5) + 200 R = 76(5)
C = 180 + 200 R = 380
C = 380
Answer:
The breakeven point is 5 units
Step-by-step explanation:
C is the cost ($) and R is the revenue ($).
We are told that :
R=76x
C=36x+200
Profit would be the Revenue minus the Cost:
Profit = R - C
The "breakeven point" is the situation in which there is neither a profit nor a loss. That means that the total profit equals the total cost for x units. We want to find the value of x that would result in no profit or loss: Profit = 0.
0 = R-C
====
Profit = 76x - (36x+200)
Profit = 40x - 200
0 = 40x - 200 [The Breakeven Point]
-40x = -200
x = 5 units
5 units results in 76*5 or $380 in revenue.
5 units results in a cost of 36*(5)+200 or $380 in costs.
The result is breakeven (Profit = $390 - $380) of $0.00
Answer!! What is Simplified?
On solving the provided question, we can say that Volume of cuboid = 2x(3x+4)(3x-4)
Volume = [tex]18x^3 + 48x^2 + 32x[/tex]
what is cuboid?A cuboid is a solid or three-dimensional form in geometry. A cuboid is a convex polyhedron having 8 vertices, 12 sides, and 6 rectangular faces. Another name for a cuboid is a cuboid. A cube is a cuboid with six square faces. Boxes include things like books and bricks. The following are the key variations between cubic and cubic: In contrast to a cube, which has rectangular faces, a cube has six equally sized square faces. Even though the cube and cuboid structures have a similar appearance, they differ in some ways in terms of side length, diagonal, and area.
here,
we have a cuboid
Volume of cuboid = 2x(3x+4)(3x-4)
Volume = [tex]18x^3 + 48x^2 + 32x[/tex]
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Triangle A”B”C” is formed using the translation (x+2, y+0) and the dilation by a scale factor of 1/2 from the origin. Which equation explains the relationship between line AB and line A”B”?
Point A: (-3,3)
Point B: (1,-3)
Point C: (-3,-3)
Answer options:
A) Line A”B” equals Line AB divided by two
B) Line AB equals Line A”B” divided by two
C) Line AB over Line A”B” equals 1/2
D) Line A”B” over Line AB equals two
Triangle ABC is dilated by a scale factor of 1/2, the line A”B” equals Line AB divided by two
What is an equation?An equation is an expression composed of variables and numbers linked together by mathematical operations.
Translation is the movement of a point up, down, left or right in the coordinate plane.
Dilation is the increase or decrease in the size of a figure.
Triangle A”B”C” is formed using the translation (x+2, y+0) and the dilation by a scale factor of 1/2 from the origin.
Since there is a dilation by a scale factor of 1/2, hence:
A"B" = AB * 1/2
A"B" = AB/2
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Re-write the quadratic function below in Standard Form y= -3(x+2)(x-3)
Answer:
-3x^2 +3x +18
Step-by-step explanation:
Y= -3(x+2)(x-3)
= - 3 (x^2 - x - 16)
= - 3x^2 - 3(-x) - 3(-6)
= - 3x^2 +3x +18
the admision to a local fair is $10 each for adult and $6 for each child. Each ride costs $1.50 for an adult and $1 for a child.
An adult and child will each go on 7 ride. How much more did the adult spend?
Answer:
7.5
Step-by-step explanation:
10 + ( 7 × 1.5) = 20.5
6 + ( 7 × 13 ) = 13
20.5 - 13 = 7.5
Ade and Ola share a sum of 4000.00 in the ratio of 3:5 find the amount Ade received solutions
The number of shares that Ade and Ola possess will be 1500 and 2500, respectively.
What is the solution to the equation?The allocation of weights to the important variables that produce the calculation's optimum is referred to as a direct consequence.
Ade and Ola share a sum of 4000.00 in a ratio of 3:5.
Let 'x' and 'y' be the number of shares that Ade and Ola have, respectively. Then the equations are given as,
x / y = 3 / 5 ...1
x + y = 4000 ...2
From equations 1 and 2, then we have
(3/5)y + y = 4000
(8/5) y = 4000
y = 2500
Then the value of the variable 'x' is calculated as,
x + 2500 = 4000
x = 1500
The number of shares that Ade and Ola possess will be 1500 and 2500, respectively.
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(16-8x-7x^2+2x^3)/(x-4)
The division of 16 - 8x - 7x² + 2x³ by x - 4 will have a quotient of 2x² + x - 4 and a remainder of 0 using synthetic division.
Dividing with synthetic divisionThe procedure for synthetic division involves the following steps:
Divide.
Multiply.
Subtract.
Bring down the next term, and
Repeat the process to get zero or arrive at a remainder.
We shall rearrange 16 - 8x - 7x² + 2x³ to become 2x³ - 7x² - 8x + 16 and then divide by x - 4 as follows;
2x³ divided by x equals 2x²
x - 4 multiplied by 2x² equals 2x³ - 8x²
subtract 2x³ - 8x² from 2x³ - 7x² - 8x + 16 will give x² - 8x
x² divided by x equals x
x - 4 multiplied by x equals x² - 4x
subtract x² - 4x from x² - 8x will give us -4x + 16
-4x divided by x equals -4
x - 4 multiplied by -4 equals -4x + 16
subtract -4x + 16 from -4x + 16 will result to a remainder of 0
Therefore by synthetic division, 16 - 8x - 7x² + 2x³ divided by x - 4 is equal to the quotient of 2x² + x - 4 with a remainder of 0.
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In a lab experiment, a population of 50 bacteria is able to quadruple every hour. Which equation matches the number of bacteria in the population after 3 hours?
The exponential equation that matches the number of bacteria after 3 hours is given as follows:
y = 50(4)^3.
What is an exponential function?An exponential function is used to model this situation as the output is multiplied for each unit of the input.
The general format of an exponential function is given as follows:
y = a(b)^x.
In which:
a represents the initial value.b represents the rate of change.In a lab experiment, a population of 50 bacteria is able to quadruple every hour, hence the parameters a and b are given as follows:
a = 50, b = 4.
Thus the exponential function that gives the population after x hours is of:
y = 50(4)^x.
After three hours, we have that x = 3, hence the population is given as follows:
y = 50(4)^3.
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What does to the power of negative 1 mean?
positive number to the power negative 1 is a number that is always less than 1.
Explanation: a-m = 1/am. when calculate have the result when a positive number is raised to the power of -1, we always get a number that is less than one; for example, 2-1 = 1/2 < 1.
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( x - 8 )( x + 8)
this is Operation Multiplication btw
For the expression (x - 8)(x + 8), using the arithmetic operation the value for x is x = 8.
What is arithmetic operation?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
The given expression is - (x - 8)(x + 8)
Use the arithmetic operation of multiplication -
(x - 8)(x + 8)
x(x + 8) - 8(x + 8)
x² + 8x - 8x -64
Now, use the arithmetic operation of addition and subtraction -
x² + 8x - 8x -64
x² - 64
Now, collect the like terms on each side -
x² = 64
x = √64
Now, solve the radical form to get the value of x -
x = √64
x = 8
Therefore, the value of x is obtained as x = 8.
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What is the coefficient of the variable in the expression 2 − 5x − 4 8?
So, on solving the provided question, we can say that coefficient of the variable in the expression 2 − 5x − 4 8 is 5
What is the coefficient?A coefficient in mathematics is a polynomial, a series, or the multiplicative coefficient of a particular term in an expression. Typically numeric, however any expression is permitted. The term "parameter" can also refer to the coefficients themselves if they are variables. A number times a variable equals a coefficient. Coefficient examples include: The coefficient is 14 in phrase 14c 14c 14c. The coefficient is 1 for word g. multiplies the variable by this amount. example Given that "z" is a variable and that 6z is the definition of the term, the coefficient is 6. One is the coefficient of the square of x.
here,
coefficient of the variable in the expression 2 − 5x − 4 8 is 5
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Gustave imagen es Space And Time And The for him steps Make Up one mile one day gusta e walked 6300 steps which one was 3 miles
Based on the information, it can be inferred that Gustave gives 2,100 steps for every mile he walks.
How to calculate the number of steps Gustave takes in a mile?To calculate the number of steps Gustave takes in a mile, we must take into account the information available about the ratio of steps to the distance he covers. In this case, we know that for every 3 miles, Gustave takes 6,300 steps.
To find how many steps he takes in a mile we must divide the number of steps into the number of miles he walked as shown below:
6,300 / 3 = 2,100According to the above, Gustave takes 2,100 steps for every mile he walks.
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write the equation of a horizontal line that goes through the point (3,7). Explain how you got your answer. I will mark you brainliest!
Answer:
y = 7
Step-by-step explanation:
The equation of a horizontal line is worth memorizing. It is:
y = anumber
(and just in case for future reference the equation of a vertical line is
x= anumber)
Well, you are given one point (3,7).That is in the form (x,y). So x is 3 and y is 7.
y is 7
y = 7
(if it helps...horizontal lines have 0 slope. So m = 0. The equation in point-intercept form is
y = mx + b
y = 0x + b
y = b this is the y= anumber) originally mentioned here)
How do you solve equations of equality?
Answer:
If two expressions are equal to each other, and you add the same value to both sides of the equation, the equation will remain equal. When you solve an equation, you find the value of the variable that makes the equation true. In order to solve the equation, you isolate the variable.
Step-by-step explanation: