TT is equal to T-1, so the given matrix is an orthogonal matrix.
What is an orthogonal matrix?
A matrix is said to be orthogonal if its antipode is also its transpose and if its relationship to the inner product is established. A real square matrix with orthonormal vectors in its columns and rows is known as an orthogonal matrix or orthonormal matrix in direct algebra. Where QT is the transpose of Q and I is the identity matrix is one system to put this into words. However, the matrix is said to be orthogonal, If the transpose of a square matrix with real figures or values is equal to the inverse matrix of the matrix. In other words, an identity matrix will always be affected by the product of a square orthogonal matrix and its transpose.Here, the determinant of T,
|T| = cos²∅ + sin²∅
= 1
Now, its cofactors,
[tex]T_{11}[/tex] = cos ∅
[tex]T_{12[/tex] = sin ∅
[tex]T_{21[/tex]= -sin ∅
[tex]T_{22[/tex] = cos ∅
Now, the adjoint matrix of cofactors matrix transpose [tex]T^T[/tex] is equal to [tex]T^{-1.[/tex]
Hence, the given matrix is an orthogonal matrix.
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Solve the system of equations using elimination: -6x-6y=-42−6x−6y=−42 and -x+2y=8−x+2y=8.
The solution of the equations −6x − 6y = −42 and −x + 2y = 8 will be (2, 5).
What is the solution to the equation?The distribution of weights to the variables involved that establishes the equilibrium in the calculation is referred to as a result.
The equations are given below.
−6x − 6y = −42 ...1
−x + 2y = 8 ...2
From equation 2, we have
x = 2y − 8
Put the value of x in equation 1, then we have
−6(2y − 8) − 6y = −42
−12y + 48 − 6y = − 42
18y = 90
y = 5
Then the value of x will be
x = 2(5) − 8
x = 10 − 8
x = 2
The solution of the equations −6x − 6y = −42 and −x + 2y = 8 will be (2, 5).
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There are boys and girls in a class in the ratio 19:6 There are 104 more boys than girls. How many boys are there?
There are 152 boys who are present in the class with ratio 19:6.
What is ratio?A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.
Given:
ratio = 19:6
let the common ratio be x
number of boys = 19 x
number of girls = 6x
Also,
number of boys = number of girls + 104
19x = 6x +104
13 x = 104
x= 8
Hence, the number of boys are 19x= 19*8 = 152 boys.
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(Dividing Rational Numbers MC)
(Dividing Rational Numbers MC)
Solve 0.504 ÷ 8.
−7.496
0.063
0.630
4.032
Solve 0.504 ÷ 8.
−7.496
0.063
0.630
4.032
The correct answer for 0.504 ÷ 8 is option B which is 0.063
How to solve 0.504 ÷ 80.504 / 8 is solved by removing the decimals first and then the division as required.
0.504 / 8
= 504/1000 / 8
= 504 / 8000
= ( 504 / 8 ) * ( 1 / 1000 )
= 63 * ( 1 / 1000 )
= 0.063
What are rational numbers?These are class of numbers that can be represented as fractions inform of two integers. Such numbers while in fractions, the denominator should not be zero and while decimal do not need to approximations to terminate them.
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Exact and Approximate Pi
Answer: c = 12
Step-by-step explanation: r is 2 since in the circle it shows the length from the center to the arc. now multiply 2(3)(2) and you will end up with 12.
Answer:
Circumference = 12
Step-by-step explanation:
C=2πr
where,
π = 3r = 2➟ C = 2πr
Insert values
➟ C = (2)(3)(2)
➟ C = (6)(2)
➟ C = 12
what value of c is in the solution set of
2(3x-1)>4x-6
Answer:
x> -2
Step-by-step explanation:
17 Select the correct answer.
What are the solutions of this quadratic equation?
4x² + 3 = 4x + 2
OA. x = 1/ 2
OB. x = -2 ± √3
OC. x = 2
OD. x = 1+√3
Find the measures of the angle, its complement, and its suplement if three times the measure of a suplement of an angle is eight times the measure of a complement of the angle.
The angle, its complement, and its supplement are 36, 54 and 144 degrees respectively
Supplementary and complementary anglesThe sum of complementary angles s 90 degrees while that of supplementary angles is 180 degrees.
Let the complement of an angle be 90 -x and its supplement be 180 -x
If three times the measure of a supplement of an angle is eight times the measure of a complement of the angle, then;
3(180-x) = 8(90-x)
Expand
540-3x = 730 - 8x
540-720 = -8x + 3x
180 = 5x
x = 180/5
x = 36 degrees
Complement = 90 - 36 = 54degrees
Supplement = 180 - 36 = 144 degrees
Hence the angle, its complement, and its supplement are 36, 54 and 144 degrees respectively
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for the given set, first calculate the number of subsets for the set, then calculate the number of proper subsets. { 5, 7, 17, 15 }
Subsets of the specified set equal 16.
The set's appropriate subset is equal to 15.
What is set known as?A set is the collection of specific, distinct items from our perception (Anschauung) and from our thought, which are referred to as the set's elements. A set's elements or members can be anything, including numbers, people, alphabet letters, other sets, and so on. Sets are typically indicated.
According to the given data:given data:- { 5, 7, 17, 15 }
this set have 4 elements:
We are aware that the number of subsets of a set with n elements is represented by 2^n
where n denotes the set's element count.
as a result, the given set's subsets = 2^4
= 16
Because the original set itself is subset of itself, it is not a legitimate subset.
Consequently, a proper subset of a set:- 2^n - 1
= 2^4 - 1
= 16 -1
= 15
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what Is the value of 343 ⅓ ?
Answer:
7. I attached my answer ... have a nice day
(a) A pyramid is made above a cuboid with measure 90 cm x 50 cm × 225 cm. If the height of the pyramid including cuboid is 240 cm, find the total volume.
The total volume of the solid = 1,035,000 cm³
What is the Volume of a Pyramid and a Cuboid?Volume of a Pyramid = 1/3(length)(width)(height).
Volume of a cuboid = (length)(width)(height).
Volume of the Pyramid = 1/3(length)(width)(height)
Volume of the Pyramid = 1/3(90)(50)(240 - 225)
Volume of the Pyramid = 1/3(90)(50)(15)
Volume of the Pyramid = 22,500 cm³
Volume of the cuboid = (length)(width)(height)
Volume of the cuboid = (90)(50)(225)
Volume of the cuboid = 1,012,500 cm³
The total volume of the solid = 22,500 + 1,012,500
The total volume of the solid = 1,035,000 cm³
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The average income,I, in dollars, of a lawyer with an age of x years is modeled with the following function:
Solving a quadratic equation, the youngest age for which the average income of the lawyer is of $250,000 is of 26 years.
What is a quadratic function?A quadratic function is given according to the following rule:
y = ax² + bx + c
[tex]\Delta = b^2 - 4ac[/tex]
The solutions are:
[tex]x_1 = \frac{-b + \sqrt{\Delta}}{2a}[/tex]
[tex]x_2 = \frac{-b - \sqrt{\Delta}}{2a}[/tex]
In which:
[tex]\Delta = b^2 - 4ac[/tex]
For this problem, the equation that models the income as function of age is given by:
I(x) = -425x² + 45500x - 650000.
The income is of 250,000 when I(x) = 250000, hence:
250000 = -425x² + 45500x - 650000.
425x² - 45500x + 900000 = 0.
Hence the coefficients are a = 425, b = -45500 and c = 900000, then:
[tex]\Delta = (-45500)^2 - 4(425)(900000) = 540250000[/tex][tex]x_1 = \frac{45500 + \sqrt{540250000}}{2(425)} = 80.87[/tex][tex]x_2 = \frac{45500 - \sqrt{540250000}}{2(425)} = 26.18[/tex]The youngest age for which the average income of the lawyer is of $250,000 is of 26 years. (the oldest is of 81 years).
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A line has a slope of 1 and passes through the point (-5, 4). What is its equation in
slope-intercept form?
Answer:
y = x + 9
Step-by-step explanation:
y = mx + b Use the x and y from the point (-5,4) and the slope to find b. They you can write the equation. An ordered pair is in the form of (x,y)
y = mx + b
4 = (1)(-5) + b
4 = -5 + b Add 5 to both sides of the equation
9 = b
Now write the equation
y = x + 9
Delila is multiplying three negative numbers and determines that the solution will be negative . Is she correct or incorrect? Justify your thinking.
Answer: Delila is correct about this determination.
Step-by-step explanation: The rule for multiplying two negatives is that a negative and a negative always equal a positive. Therefore, multiplying two negatives will give a positive. However, we still have to multiply that positive by another negative to get our answer. Another multiplication rule states a negative and a positive always equals a negative. Let's set up an example with the numbers -7, -8, and -4:
Using the first multiplication rule mentioned above, let's multiply:
-8 x -7 = 56
Using the second multiplication rule mentioned above, we can finish the equation and get our answer:
56 x (-4) = -224
-7 x -8 x -4 = -224.
Final answer: Delila is correct. Multiplying three negatives will result in a negative solution.
2 { 5 5/8 } + 3 { 9 13/16 } ?
Help Please
The equivalent expression for the expression consisting of mixed fraction 2 { 5 5/8 } + 3 { 9 13/16 } is 651/16.
According to the given question.
We have an expression consisting of mixed fractions.
2 { 5 5/8 } + 3 { 9 13/16 }
Since, we convert the mixed fraction into improper fraction by
Step 1: Multiply the denominator with the whole number.
Step 2: Add the numerator to the product obtained from step 1.
Step 3: Write the mixed fraction with the sum obtained from step 2 as the numerator and the denominator of the original fractional part of the mixed fraction.
Therefore, the equivalent expression for the expression 2 { 5 5/8 } + 3 { 9 13/16 } is given by
2{45/8} +3{157/16}
= 90/8 + 471/16
= [180 + 471]/16
= 651/16
Hence, the equivalent expression for the expression consisting of mixed fraction 2 { 5 5/8 } + 3 { 9 13/16 } is 651/16.
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Simplify the following
Need the answer asap
answer:
1.
[tex]9y - 1[/tex]
2.
[tex] { \times }^{3} - { \times }^{2} + \times [/tex]
3.
[tex] - 3 {a}^{2} + a + 2[/tex]
4.
[tex] - { \times }^{2} [/tex]
5.
[tex] - 4 { \times }^{5} + 16 { \times }^{3} - 20 { \times }^{2} [/tex]
6.
[tex]8 { \times }^{3} + 6 { \times }^{2} - 32x + 15[/tex]
7.
[tex] { \times }^{2} - \times - 20[/tex]
8.
[tex]24 { \times }^{2} - 6 \times + 1[/tex]
9.
[tex] { \times }^{2} - 6 \times + 5[/tex]
10.
[tex]10 { \times }^{5} + 3 { \times }^{3} - 2 \times [/tex]
what is 22.652 rounded to the nearest tenth?
Answer:
22.7
This is becuase 6 is closer too 7. Thus we would round too 7.
If m ∠ E F G = 35 ° and m ∠ G F H = 63 ° , what is m ∠ E F H ?
Answer:
28°
Step-by-step explanation:
63 -35 = 28
In a certain chemical, the ratio of zinc to copper is 4 to 19 . A jar of the chemical contains 551 grams of copper. How many grams of zinc does it contain?
Answer:
116 g
Step-by-step explanation:
Zn:Cu = 4:19
so 551 x (4/19) = 116
simplify -(6x + 22) - 8(3 - x)
a. In Euclidean geometry, a line continues in opposite directions forever. As a result, the line will have an infinite length. How does this differ from a line in spherical geometry?
Euclidean geometry differs from spherical geometry because Euclidean geometry is used to set points by the use of plane.
What is Euclidean geometry?Euclidean geometry is the study of geometrical shapes (plane and solid) and figures based on different axioms and theorems.
Now since Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sphere" are used for the surface together with its 3-dimensional interior.
Euclidean geometry differs from spherical geometry because Euclidean Geometry are known to make use of a plane to be able to set points and lines, but Spherical Geometry are known to make use of spheres to set up points and great circles.
Thus, Euclidean geometry differs from spherical geometry because Euclidean geometry is used to set points by the use of plane.
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15 women can complete a piece of work in 16 days. How many women
should leave so that the work would be finished in 20 days?
Determine whether each relationship is linear or exponential. Then write an equation to represent the relationship.
5. The relationship is Linear and the equation is 500 × 1.1 + 500 = 1050.
6. The relation ship is Linear and the equation is 6 × 2/3 = 4
Given,
5. The population of a town = 500
The growth of population expected by the city planner = 1.1
That is, 500 × 1.1 + 500 = 1050
This is a Linear equation.
6. The distance from where the basketball fall down = 6 feet
Basket ball bounce back its previous height = 2/3
That is, 6 × 2/3 = 4
This is a Linear equation.
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Christy buys 3 1/4 pounds of pork roast and 5 1/2 pounds of beef roast. She pays a total of $67.50. The pork costs $0.50 per pound less than the beef. What is the combined cost of one pound of pork and one pound of beef?
The combined cost of one pound of pork and one pound of beef is $ 15.5.
How to estimate the combine cost of two articles bought in a butchery
The total cost is equal to the sum of the products of unit price and quantity of an article. In this problem we find the sum of the quantities of two products (pork roast, beef roast), whose expression can be described below:
67.50 = (3.25) · (x - 0.50) + (5.5) · x
Where x is the unit cost of the beef roast.
Now we proceed to solve the equation for x:
67.50 = 3.25 · x - 1.625 + 5.5 · x
69.125 = 8.75 · x
x = 7.9
A pound of beef roast costs $ 7.9 and a pound of pork roast costs $ 7.4. Then, the combined cost of one pound of prok and one pound of beef is:
C = 1 · 7.9 + 1 · 7.4
C = 15.5
The combined cost of one pound of pork and one pound of beef is $ 15.5.
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Find 3/4ab divided by 6/abc. Write the quotient in simplest form.
Answer:
[tex] \sf \large \frac {1}{8}b²c[/tex]
Step-by-step explanation:
3/4ab/6/abc
= 1/8ab²c/a
= 1/8b²c
just need help on 17-18! homework is due tonight! ill give brainliest!
Answer: 17. The fower will last 85 days
18. She ran at a constant pace of around 12.94mph (if her time was 14 minutes and 24.6 (14.14m) ) or 12.7mph (if her time was 14 minutes 41 seconds) (the .41 confused me so I did both possible solutions)
Hope this helped c:
A group of friends wants to go to the amusement park. They have $235.75 to spend on parking and admission. Parking is $6.25, and tickets cost $25.50 per person, including tax. Write and solve an equation which can be used to determine pp, the number of people who can go to the amusement park.
Answer:
The equation for the number of people is 25.50x + 6.25 = 235.75 and the number of people who can go to the amusement park are 9.
Step-by-step explanation:
We have,
Spend on parking and admission = $235.75
Spend on parking = $6.25
Cost of ticket per person = $25.50
Let number of number of people = x
Now,
According to the question;
25.50x + 6.25 = 235.75
So, this the equation for number of number of people.
Now,
25.50x + 6.25 = 235.75
25.50x = 229.50
x = 9
So, number of number of people is 9, which we find out using the above mentioned equation.
Hence, we can say that the equation for the number of people is 25.50x + 6.25 = 235.75 and the number of people who can go to the amusement park are 9.
The number of people who can go to the amusement park is 9 they have $235.75 to spend on parking and admission. Parking is $6.25.
What is a linear equation?It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.
If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.
It is given that:
A group of friends wants to go to the amusement park. They have $235.75 to spend on parking and admission.
Parking is $6.25, and tickets cost $25.50 per person, including tax.
Let x be the number of people
6.25 + 25.50x = 235.75
25.50x = 235.75 - 6.25
25.50x = 229.5
x = 9
Thus, the number of people who can go to the amusement park is 9 they have $235.75 to spend on parking and admission. Parking is $6.25.
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f(x) = 3x + 4 and g(x) = 2x - 3, find (f - g)(x)
2
Step-by-step explanation:
8(5-6v)=-7-v
help please :))
Answer:
V = 1 (Please mark the answer as brainliest)
Step-by-step explanation:
8(5 - 6v) = -7 - v
40 - 48v = -7 - v
40 = -7 + 47v
47 = 47v
1 = v
Answer:
v = 1
Step-by-step explanation:
Let's solve the problem,
→ 8(5 - 6v) = -7 - v
→ 40 - 48v = -7 - v
→ 40 + 7 = -v + 48v
→ 47v = 47
→ v = 47/47
→ [ v = 1 ]
Hence, the value of v is 1.
Keitaro walks at a pace of 3 miles per hour and runs at a pace of 6 miles per hour. Each month, he wants to complete at least 36 miles but not more than 90 miles. The system of inequalities represents the number of hours he can walk, w, and the number of hours he can run, r, to reach his goal.
3w + 6r ≥ 36
3w + 6r ≤ 90
A graph shows w on the x-axis, from 0 to 30, and t on the y-axis, from 0 to 24. Two solid lines are shown. The first line has a negative slope and goes through (0, 6) and (12, 0). Everything above and to the right of the line is shaded. The second line has a negative slope and goes through (0, 15) and (30, 0). Everything to the left of the line is shaded.
Which combination of hours can Keitaro walk and run in a month to reach his goal?
2 hours walking; 12 hours running
4 hours walking; 3 hours running
9 hours walking; 12 hours running
12 hours walking; 10 hours running
The combination of hours that Keitaro can walk and run in a month to reach his goal will be A. 2 hours walking; 12 hours running
How to illustrate the information?From the information, Keitaro walks at a pace of 3 miles per hour and runs at a pace of 6 miles per hour. Each month, he wants to complete at least 36 miles but not more than 90 miles.
The equation is
3w + 6r ≥ 36
3w + 6r ≤ 90
The combination of hours that Keitaro can walk and run in a month to reach his goal will be 2 hours walking, and 12 hours running.
This will be:
3w + 6r ≤ 90
3(2) + 6(12)
6 + 72 = 78
Therefore, 78 is less than 90.
The correct option is A.
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henery works at a pet shelter after school. he purchases a large package of dog treats. he sets aside 10 treats and distributes the rest equally among the 15 dogs in the shelter. if each dog received 4 treats, write and solve an equation to find the number of treats t that were in the original package
An equation to find the number of treats t that was in the original package is t=10+4(15).
First, you set aside the 10 and that is the start of your equation.
We need to find an equation to find the number of treats t that was in the original package.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Now, t=10+4(15)
⇒t=10+60
⇒t=70
Therefore, an equation to find the number of treats t that was in the original package is t=10+4(15).
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