We have verified that A^-1A = I and AA^-1 = I for both matrices.
To compute the inverse of a matrix, we can use the formula: A^-1 = 1/det(A) * adj(A), where det(A) is the determinant of the matrix A and adj(A) is the adjoint of the matrix A.
(a) A = [1 1; 1 -4]
det(A) = (1*-4) - (1*1) = -4 - 1 = -5
adj(A) = [-4 -1; -1 1]
A^-1 = 1/-5 * [-4 -1; -1 1] = [4/5 1/5; 1/5 -1/5]
(b) A = [2 1 1; 3 1 2; -1 3 -1]
det(A) = (2*(1*-1) - 1*(2*3) + 1*(-1*3)) - (1*(1*-1) - 1*(2*-1) + 1*(3*3)) = -6 - 3 - 3 - 9 = -21
adj(A) = [-1 -2 7; 5 1 -8; 2 5 -7]
A^-1 = 1/-21 * [-1 -2 7; 5 1 -8; 2 5 -7] = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21]
To verify that A^-1A = I and AA^-1 = I, we can simply multiply the matrices and check if the result is the identity matrix.
For (a):
A^-1A = [4/5 1/5; 1/5 -1/5] * [1 1; 1 -4] = [1 0; 0 1] = I
AA^-1 = [1 1; 1 -4] * [4/5 1/5; 1/5 -1/5] = [1 0; 0 1] = I
For (b):
A^-1A = [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] * [2 1 1; 3 1 2; -1 3 -1] = [1 0 0; 0 1 0; 0 0 1] = I
AA^-1 = [2 1 1; 3 1 2; -1 3 -1] * [1/21 2/21 -7/21; -5/21 -1/21 8/21; -2/21 -5/21 7/21] = [1 0 0; 0 1 0; 0 0 1] = I
Therefore, we have verified that A^-1A = I and AA^-1 = I for both matrices.
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Marquis and Yolanda plan to sell T-shirts at their school’s Community Day. They make 25 shirts and each costs $15 to make. If they would like to make $320 in profit, how much should they sell each T-shirt for?
They should sell each T-shirt for $27.80 to make a profit of $320.
Describe Profit?Profit is the financial gain that a company or individual makes after deducting all the expenses from the revenue earned. It is the positive difference between the total revenue generated from the sale of goods or services and the total cost of producing and selling those goods or services.
Profit is one of the key indicators of a company's financial performance, and it is calculated by subtracting the total expenses from the total revenue. The expenses include the cost of raw materials, labor, rent, taxes, and other overhead costs associated with producing and selling the goods or services.
To make a profit of $320, they need to sell all 25 shirts for a total of $320 + ($15 x 25) = $695.
Therefore, the price they need to sell each T-shirt for is $695 ÷ 25 = $27.80 (rounded to the nearest cent).
So they should sell each T-shirt for $27.80 to make a profit of $320.
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What is the area of the circle? Approximate using pi equals 22 over 7 and round to the nearest square yard.
a circle with radius labeled 18 yards
57 square yards
113 square yards
1,018 square yards
2,037 square yards
Answer: A
Area of a circle=[tex]\pi r^2\\[/tex]
Now that we have established this, we will now substitute the values in the equation.
[tex]\pi r^2\\=\frac{22}{7}\\ =\frac{396}{7}\\ =56.57 \\[/tex] OR 57 square yards
HOPE THIS HELPED PLEASE MARK BRAINLIEST IF IT DID!
Answer:
If your options are 110 square yards
220 square yards
3,850 square yards
15,400 square yards
then the correct answer is 3,850 square yards.
Step-by-step explanation:
I took the quiz and got it right.
Nine children want to feed the birds they eash have 2 bags of seed. How many bags of seed are there in all
Answer: 18
Step-by-step explanation:
9 kids each have two bags
9 x 2 = 18
9 + 9 = 18
A constant force of 18 newtons is being applied at a constant angle of 56° on an object at the same time that a constant force of 32 newtons at
a constant angle of 124° is acting on the object. What is the magnitude and direction of the resultant force?
The magnitude and direction of the resultant force will be 7.829 N and the direction is opposite to the applied force.
What is a vector?A vector is an amount that has direction as well as magnitude and complies with the rule of vector addition.
A constant force of 18 newtons is supplied to an item at a constant angle of 56° while a constant force of 32 newtons is produced to the entity at a fixed angle of 124°.
Then the magnitude and direction of the resultant force are given as,
∑F = 0
∑F = 18 · cos 56° + 32 · cos 124°
∑F = 10.065 - 17.894
∑F = - 7.829 N
The magnitude and direction of the resultant force will be 7.829 N and the direction is opposite to the applied force.
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PLEASE HELP
what is 7.5% of 48.30? show your working out pls.
Answer:
3.6225
Step-by-step explanation:
7.5% = 0.075
48.30 x 0.075 = 3.6225
So, 7.5% of 48.30 is 3.6225
A school staff meeting had 30 teachers in attendance, 30% of whom were first-year teachers. How many first-year teachers were in the meeting?
Answer:
There were 30 x 30/100 = <<30*30/100=9>>9 first-year teachers at the meeting.
Answer: 9.
Answer:
9
Step-by-step explanation:
30% of 30 is 9. Therefore there are 9 teachers how were first year teachers.
Hope this helps :)
V varies partly as D and partly as the square of D when Vequals to 5 ,D equals to 2, and when V equal to 9 and D equals to 3
write this law connecting V and D
[tex]\stackrel{ \textit{Partial Variation} }{V=aD+bD^2}\qquad \impliedby \begin{array}{llll} \textit{V\textit{ varies partly}}\\ \textit{with D and partly with }D^2 \end{array} \\\\[-0.35em] ~\dotfill\\\\ \textit{we also know that} \begin{cases} V=5\\ D=2\\[-0.5em] \hrulefill\\ V=9\\ D=3 \end{cases}\implies \begin{array}{llll} 5=a2+b2^2&\qquad &5=2a+4b\\\\ 9=a3+b3^2&&9=3a+9b \end{array} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{using the 1st equation}}{5=2a+4b}\implies 5-2a=4b\implies \cfrac{5-2a}{4}=b \\\\\\ \stackrel{\textit{using the 2nd equation}}{9=3a+9b}\implies \stackrel{\textit{substituting from above}}{9=3a+9\left( \cfrac{5-2a}{4} \right)}\implies 9=3a+\cfrac{45-18a}{4} \\\\\\ \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{4}}{4(9)=4\left( 3a+\cfrac{45-18a}{4} \right)}\implies 36=12a+45-18a\implies -9=-6a[/tex]
[tex]\cfrac{-9}{-6}=a\implies \boxed{\cfrac{3}{2}=a}\hspace{5em}b=\cfrac{5-2\left( \frac{3}{2} \right)}{4}\implies \boxed{b=\cfrac{1}{2}} \\\\[-0.35em] ~\dotfill\\\\ ~\hfill V=\cfrac{3}{2}D+\cfrac{1}{2}D^2~\hfill[/tex]
How many possible rational roots does the polynomial 6x^(4) - 11x^(3) + 8x^(2) - 33x - 30 have?
The possible rational roots are -1/6, -1/3, -1/2, -1, 1/6, 1/3, 1/2, 1 .
The polynomial 6x4 - 11x3 + 8x2 - 33x - 30 has 4 possible rational roots. To find them, use the Rational Root Theorem.
This theorem states that any rational roots of a polynomial must be in the form a/b, where a is a factor of the constant term and b is a factor of the leading coefficient.
In this case, the constant term is -30, so the possible factors are -1, -2, -3, -5, -6, -10, -15, -30. The leading coefficient is 6, so the possible factors are 1, 2, 3, 6.
Therefore, the possible rational roots are -1/6, -1/3, -1/2, -1, 1/6, 1/3, 1/2, 1.
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Find the area of each shaded region.
Milo sets sail from a dock and heads in a straight line for 7 miles. The wind then picks up momentarily and he is forced to change direction by 5pi/36. He then sails in a straight line in this new direction for another 4 miles. At this point, how far is he from the dock? Round to 2 decimal places.
Milo is 8.09 miles from the dock.
Here we can use the Law of Cosines,
which states that c² = a² + b² - 2abcos(C),
Where c is the side opposite the angle C in a triangle.
Call the distance from the dock to the point where Milo changes direction "a", the distance Milo sails in the new direction "b", and the angle between these two sides "C".
We know that a = 7 miles and b = 4 miles.
To find C, we can use the fact that Milo changes direction by 5π/36 radians.
Since he turned to the right, we can say that he turned by,
⇒ 360 - 5π/36 = 355π/36 radians.
This is the angle between the two sides of the triangle.
Now we can plug these values into the Law of Cosines to find the distance from Milo to the dock:
⇒ c² = a² + b² - 2abcos(C)
⇒ c² = 7² + 4² - 2(7)(4)cos(355π/36)
⇒ c² = 49 + 16 - 56cos(355π/36)
⇒ c² = 65.47
Taking the square root of both sides, we get:
c = 8.09 miles (rounded to 2 decimal places)
Therefore, Milo is 8.09 miles from the dock.
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increase 72 in the ratio 3:5
Answer:
75:77
Step-by-step explanation:
5 is 2 more than 3
3+72=75
5+72=77
therefore: 3:5 = 75:77
If that's not what you meant, then I don't understand the question
Kate buys a greeting card for 3.79. She then buys 4 postcards that all cost the same amount. The total cost is 5.11. How much is each postcard? Show your work.
Answer:
$0.33
Step-by-step explanation:
5.11-3.79=1.32÷4=.33
i need y'all help on dis
Answer:
Customer A - 5 / 22.45 = $0.22 or 22 cents per quart
Customer B - 7 / 25.45 = $0.27 or 27 cents per quart
The number of carbohydrates from 10 different tortilla sandwich wraps sold in a grocery store was collected. Which graphical representation would be most appropriate for the data, and why?
Circle chart, because the data is categorical
Line plot, because there is a large set of data
Histogram, because you can see each individual data point
Stem-and-leaf plot, because you can see each individual data point
Answer: Stem-and-leaf plot, because you can see each individual data point.
Step-by-step explanation:im taking the test.
Answer:stem and leaf plot
A little help here please !!
Answer:
Step-by-step explanation:
I need help guys
I need this by friday plsss
The length of the midsegment of the given trapezoid is 16.
What is a trapezoid?
A trapezoid is necessarily a convex quadrilateral in Euclidean geometry. The parallel sides are called the bases of the trapezoid.
Given is a trapezoid TUVW, with bases UV and TW given by expressions, 3x-1, 8x and the midsegment XY is 3x+7.
We need to find the length of the midsegment,
We know,
Midsegment = (base 1 + base 2) / 2
Therefore,
XY = UV+TW / 2
3x+7 = 3x-1 + 8x / 2
6x+14 = 11x-1
5x = 15
x = 3
XY = 3(3)+7 = 16
Hence, the length of the midsegment of the given trapezoid is 16.
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What is the solution for 3 1/4 + 3/8
Result in decimals: 3.625
Step-by-step explanation:
Answer:
29/8 or 3 5/8
Step-by-step explanation:
first turn 3 1/4 to 13/4
now change 13/4 to get 26/8
now add 26/8+ 3/8 and get 29/8 or 3 5/8
in a certain town there were 171 robberies last year . this year the number of robberies has gone down 39%. how many robberies were there this year , to the nearest whole number ?
The robberies that took place this year to the nearest whole number is 104 robberies.
What distinguishes a percent reduction from a percent difference?A value's percentage change from its initial value is expressed as a percent reduction, but the percentage difference between two values is expressed as a % change from their average.
Let us suppose the number of robberies this year = x.
Given that,
This year the number of robberies has gone down 39%.
x = 171 (1 - 39/100)
x = 171 (1 - 0.39)
x = 104.31
Hence, the robberies that took place this year is 104 robberies.
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What’s the value of -√22
Answer:
the answer is 22
Step-by-step explanation:
-22 x -22=22
I need the domain and range of this graph. Help quick please!
The function seen on Cartesian plane has the following domain and range:
Domain: 0 ≤ t ≤ 100, Range: 450 ≤ q ≤ 1200
How to determine the domain and the range of the function
In this problem we find a representation of a linear function on a Cartesian plane, of which we need to derive the definitions of its domain and its range. The domain corresponds to the set of times (t), in minutes, and the range is the set of amounts of water (q), in liters. The domain belongs to the horizontal axis and the range to the vertical axis.
Then, by direct inspection, the domain and the range of the function is:
Domain
0 ≤ t ≤ 100
t ∈ [0, 100]
Range
450 ≤ q ≤ 1200
q ∈ [450, 1200]
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PLSSSS HELP IF YOU TURLY KNOW THISSS
Answer:
[tex] \sf \: x = 5[/tex]
Step-by-step explanation:
Now we have to,
→ Find the required value of x.
The equation is,
→ 11 + 2x = 3(x + 2)
Then the value of x will be,
→ 11 + 2x = 3(x + 2)
→ 11 + 2x = 3x + 6
→ 2x - 3x = 6 - 11
→ -x = -5
→ [ x = 5 ]
Hence, the value of x is 5.
[tex] \: [/tex]
To find:-[tex] \textrm{the value of x = ?}[/tex][tex] \: [/tex]
Solution:-[tex] \textrm{11 + 2x = 3( x + 2 )}[/tex][tex] \: [/tex]
[tex]\textrm{11 + 2x = 3x + 6}[/tex][tex] \: [/tex]
[tex]\textrm{2x - 3x = 6 - 11}[/tex][tex] \: [/tex]
[tex]\textrm{-1x = -5}[/tex][tex] \: [/tex]
[tex]\rm{x = \cancel\frac{ - 5}{ - 1} }[/tex][tex] \: [/tex]
[tex]\underline { \boxed{\textrm{\purple{x = 5}}}}[/tex][tex] \: [/tex]
The value of x is 5 !
━━━━━━━━━━━━━━━━━━━━━━━
hope it helps! :)
Part A
By the end of its fourth week, a movie had grossed $9.2 million. Assume the revenue y in millions of dollars is
proportional to the week x.
Movie Sales
Revenue (millions of dollars)
987654321
y
X
0 1 2 3 4 5 6 7 8 9
Week
Graph the equation on your own paper. Which of the following ordered pairs does your graph pass through? Select
all that apply.
A) (4,9.2)
B) (9.2,4)
C) (1,9.2)
D) (9.2,1)
E) (0,0)
F) (1,2.3)
The ordered pair which the graph pass through include the following:
A) (4, 9.2)
E) (0, 0)
F) (1, 2.3)
How to determine the constant of proportionality?In Mathematics, a proportional relationship is a type of relationship that generates equivalent ratios and it can be modeled by the following mathematical expression:
y = kx
Where:
x represent the number of week.y represent the revenue (in millions of dollars).k represent the constant of proportionality.In order to have a proportional relationship and equivalent ratios, the variables x and y must have the same constant of proportionality:
Constant of proportionality (k) = y/x
Constant of proportionality (k) = 9.2/4
Constant of proportionality (k) = 2.3
Therefore, the required equation is given by:
y = kx
y = 2.3x
In conclusion, we would use an online graphing calculator to determine the ordered pairs that the line passes through.
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Find the missing number to create a perfect-square binomial
9x2+ X+25
Missing number to make the binomial 9x² + X + 25 a perfect square is 30x.
What is quadratic equation?Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax² + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).
Given expression
9x² + X + 25
It can be written as
(3x)² + X + 5²
Comparing with a² + 2ab + b²
a = 3x
b = 5
X = 2ab
X = 2(3x)(5)
X = 30x
Hence, 30x is the missing number to create a perfect-square binomial.
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A company began making equal deposits at the end of each quarter into an account with an APR of 6.8% compounded quarterly. They continued these deposits for six years and then stopped due to revenue problems. Four years after they stopped making deposits, the account had a value of $50,000. Find the amount deposited quarterly during the first six
years.
The company deposited $1,487.88 at the end of each quarter during the first six years.
The amount deposited quarterly during the first six years can be calculated using the formula for the future value of an annuity:
[tex]PMT = (FV / ((1 + r)^{n-1}) * (1 + r)^{-n[/tex]
where PMT is the equal quarterly deposit, FV is the final value of the account, r is the quarterly interest rate (APR / 4), and n is the number of quarters (6 years * 4 quarters per year = 24 quarters).
Plugging in the given values, we get:
[tex]PMT = ($50,000 / ((1 + 0.068/4)^{28-1}) * (1 + 0.068/4)^{-28}[/tex]
PMT = $1,487.88 (rounded to the nearest cent)
Therefore, the company deposited $1,487.88 at the end of each quarter during the first six years.
The problem involves finding the amount deposited quarterly by a company for a period of six years, given the final value of the account four years after the company stopped making deposits. This can be solved using the formula for the future value of an annuity, which calculates the total value of a series of equal payments made at regular intervals over a specified period, taking into account compound interest.
By plugging in the given values into the formula, we can solve for the equal quarterly deposit made by the company during the first six years. The resulting amount is $1,487.88, which represents the total value of all the quarterly deposits made by the company during the six-year period. This answer assumes that the interest rate remained constant throughout the entire period and that the company made no withdrawals or additional deposits after the initial six years.
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A teacher can grade 15 papers in 25 minutes. At this rate, how many papers can she grade in 90 minutes?
15/25 = x/90
25x = 1350
x = 54
54 papers
Find one solution for each equation. Assume all angles involved are acute angles. See Example 3. 31. tan a cot(a + 10°) 32. cos θ = sin(2 θ -30°) 33. sin(2θ +10°) = cos(3 θ - 20°) 34. sec (B+10°) = csc (2B + 20°) 35. tan(3B+ 4°) = cot(5B-10°) 36. cot(5 θ +2°)=tan(2 θ +4°)
By applying trigonometric function concept, it can be concluded that the solutions are:
31. tan a = cot(a + 10°), a = 40°
32. cos θ = sin(2θ - 30°), θ = 40°
33. sin(2θ + 10°) = cos(3θ - 20°), θ = 20°
34. sec(B + 10°) = csc(2B + 20°), B = 20°
35. tan(3B + 4°) = cot(5B - 10°), B = 12°
36. cot(5θ + 2°) = tan(2θ + 4°), θ = 12°
Two angles are said to be complementary angles if their sum is 90°. Sin and Cosine are complementary, Tan and Cot are complementary, and sec and cosec are complementary.
Trigonometric functions of complementary angles:
sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
tan(90° - θ) = cot θ
cot(90° - θ) = tan θ
sec(90° - θ) = csc θ
csc(90° - θ) = sec θ
31. To find one solution for the equation tan a = cot(a + 10°), we can use the fact that cot(90° - θ) = tan θ. Therefore, we can rewrite the equation as:
tan a = cot(a + 10°)
cot(90° - a) = cot(a + 10°)
90° - a = a + 10°
2a = 80°
a = 40°
32. To find one solution for the equation cos θ = sin(2θ - 30°), we can use the fact that sin(90° - θ) = cos θ. Therefore, we can rewrite the equation as:
cos θ = sin(2θ - 30°)
sin(90° - θ) = sin(2θ - 30°)
θ + 2θ - 30° = 90°
3θ = 120°
θ = 40°
33. To find one solution for the equation sin(2θ + 10°) = cos(3θ - 20°), we can use the fact that sin(90° - θ) = cos θ. Therefore, we can rewrite the equation as:
sin(2θ + 10°) = cos(3θ - 20°)
sin(2θ + 10°) = sin(90° - (3θ - 20°))
sin(2θ + 10°) = sin(110° - 3θ)
2θ + 10° = 110° - 3θ
5θ = 100°
θ = 20°
34. To find one solution for the equation sec (B + 10°) = csc(2B + 20°), we can use the fact that sec θ = csc(90° - θ). Therefore, we can rewrite the equation as:
sec(B + 10°) = csc(2B + 20°)
csc(90° - (B + 10°)) = sec(2B + 20°)
80° - B = 2B + 20°
3B = 60°
B = 20°
35. To find one solution for the equation tan(3B + 4°) = cot(5B-10°), we can use the fact that tan a = cot(90° - a). Therefore, we can rewrite the equation as:
tan(3B + 4°) = cot(5B - 10°)
cot (90° - (3B + 4°)) = cot(5B - 10°)
86° - 3B = 5B - 10°
8B = 96°
B = 12°
36. To find one solution for the equation cot(5θ + 2°) = tan(2θ + 4°), we can use the fact that cot(90° - θ) = tan θ. Therefore, we can rewrite the equation as:
cot(5θ + 2°) = tan(2θ + 4°)
tan (90° - (5θ + 2°)) = tan(2θ + 4°)
88° - 5θ = 2θ + 4°
7θ = 84
θ = 12°
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If m=log(M) and n=log(N), rewrite the following expression in terms of m and n.\( \log \left(\frac{\left(M^{\log (M)} \cdot 10^{\log (N)}\right)}{1000 \sqrt[3]{N}}\right) \)solution: m^2+2/3n-3
The expression in terms of m and n is m^2+2/3n-3.
To rewrite the given expression in terms of m and n, we need to use the properties of logarithms to simplify the expression. The properties of logarithms that we will use are:
- log(ab) = log(a) + log(b)
- log(a/b) = log(a) - log(b)
- log(a^b) = b*log(a)
Using these properties, we can rewrite the given expression as follows:
\( \log \left(\frac{\left(M^{\log (M)} \cdot 10^{\log (N)}\right)}{1000 \sqrt[3]{N}}\right) = \log \left(M^{\log (M)} \cdot 10^{\log (N)}\right) - \log \left(1000 \sqrt[3]{N}\right) \)
\( = \log \left(M^{\log (M)}\right) + \log \left(10^{\log (N)}\right) - \log \left(1000\right) - \log \left(\sqrt[3]{N}\right) \)
\( = \log (M) \cdot \log (M) + \log (N) \cdot \log (10) - \log (10^3) - \frac{1}{3} \cdot \log (N) \)
Now, we can substitute m = log(M) and n = log(N) into the expression to get:
\( = m^2 + n - 3 - \frac{1}{3}n \)
\( = m^2 + \frac{2}{3}n - 3 \)
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Direction in your noteook solve following mindline theorem show your solution
The midline or midsegment theorem calculate the value of P. the value of the variable, P that base length in the trapezoid is equals to the 13.
A trapezoid is a 4-sided (square) shape in which some sides are parallels and others not. The midsection of a trapeze is the line that runs from the middle of one leg to the middle of the other. The midsection or Midsegment theorem of a trapezium states that if a line parallel to the two bases passes through the middle of one leg, it also passes through the middle of the other leg. Also, the length of the middle fragment is half the length of two bases. Now we see a trapezoid in the image above, the length of the midsegment = 25
One of the bases of the trapezium = 37
We need to calculate the value of the other base length P Using Midsigment or Midline Theorem, Midsigment = length of two bases/2
=> 25 = (37 + P)/2
=> 50 = 37 + P
=> P = 50 - 37
=> P = 13
Hence, required length is 13.
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Complete question:
Use the midline theorem to find the value of the variable in the trapezoid. See the above figure.
Please answer I'll give you brainly
Answer:
5/8
Step-by-step explanation:
GCF:
5/8 = 25/40 ( multiply both sides by 5)
11/20 = 22/40 ( multiply both sides by 2)
please help with easy math triangle question
Answer:
not sure but i think its 12
Step-by-step explanation:
if you look carefully one side witch is 6 is have of side x and 6 x 2 is 12
hope this helps.
Answer:I think its 15 but I'm not sure
Step-by-step explanation: