The product of any two invertible matrices is invertible that is (BC) = A⁻¹
Now, let's consider the product of two invertible matrices, say A and B. Since A and B are invertible, they both have an inverse, denoted by A⁻¹ and B⁻¹, respectively.
To show that the product AB is also invertible, we need to find a matrix that satisfies the definition of an inverse. Let's call this matrix C.
If C is the inverse of AB, then we should have:
(AB)C = C(AB) = I
where I is the identity matrix.
Now, we can use the associativity of matrix multiplication to expand the left-hand side of the equation:
(AB)C = A(BC)
Notice that B and C are both matrices, so their product BC is also a matrix. Thus, we can write:
A(BC) = I
Since A is invertible, we can multiply both sides by A⁻¹ to get:
A⁻¹A(BC) = A⁻¹I
The left-hand side simplifies to:
(BC) = A⁻¹
Now, we have found a matrix C that satisfies the definition of an inverse for AB, which means that AB is invertible.
To summarize, the product of two invertible matrices is invertible, and the proof relies on the fact that the inverse of a matrix is unique and the associativity of matrix multiplication.
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What sum of money amounts to Rs 8700 in 3 years at the rate of 24% per annum?
Answer:
e. 8/00 ggg： What sum of money amounts to Rs. 8700 in 3 years at the rate of 24% per annum? [Ans: Rs. 5058.14]
Step-by-step explanation:
ABC ~ PQR. If AB : PQ = 4:5,
find A(ABC): A(PQR).
Area (ABC) is measured as Area (PQR), which equals 16:25.
To locate,
Area (ABC) is measured as: (PQR).
Solution,
This mathematical issue can easily resolved by utilising the procedure outlined below:
According to the "Area of Similar Triangles Theorem" in mathematics,
When two triangles are similar, their area ratios are proportional to the square of the ratio of the respective sides.
{Statement-1}
In light of the query and assertion 1, we can state,
Area (ABC) is measured as: (PQR)
= (AB: PQ), (BC: QR), and (AC: PR)
= (4:5)2 = (4/5)2
= 16/25 = 16:25
As a result, Area (ABC): Area (PQR) is measured at 16:25.
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The radius of a circle is 8 meters. What is the circle's circumference?
Use 3.14 for л.
Answer:
circumference=50.24
Step-by-step explanation:
c=2x3.14xr
c=2x3.14x(8)
c=50.24
1 Find the value of x.
i’m like struggling
Answer: 23 degrees
Step-by-step explanation:
Assuming that 117 is the entire angle we can find that:
94+x = 117
Subtract 94 from both sides:
x = 117-94
x = 23 degrees
What is the domain and range of the function f (x) = a superscript x? a. domain = negative real numbers, range = negative real numbers c. domain = positive real numbers, range = positive real numbers b. domain = all real numbers, range = all real numbers d. domain = real numbers, range = positive real numbers
The domain and range of the function f(x) =a^x, then option (c) Domain = positive real numbers, range = positive real numbers.
The function f(x) = a^x is an exponential function with a base of a, where a is a positive real number. The domain of the function is all real numbers, because we can raise a positive number to any real power.
However, since a is positive, a^x will always be positive, which means that the range of the function is also positive real numbers. Therefore, the correct option is c. Domain = positive real numbers, range = positive real numbers.
Therefore, the correct option is (c) Domain = positive real numbers, range = positive real numbers.
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the pictograph below shows the approximate gross revenues in the united states from four walt disney animated movies. find the ratio of the gross revenue of the hunchback of notre dame to the gross revenue of beauty and the beast. a. 1
b. 2/3
c. 3
d. 2
e. 3/2
The cost price of 20 articles is the same as sellling price of 16 articles find the gain percent
If the cost price of 20 articles is the same as selling price of 16 articles, then the gain percentage is 25%
To find the gain percent, we first need to calculate the profit earned on the sale of the 16 articles.
Let the cost price of each article be "C" and the selling price of each article be "S".
Given that the cost price of 20 articles is the same as the selling price of 16 articles, we can write:
20C = 16S
We can simplify this equation to:
S = (20/16)C = (5/4)C
Now, let's calculate the profit earned on the sale of 16 articles:
Profit = Total Selling Price - Total Cost Price
Profit = 16S - 20C
Profit = 16(5/4)C - 20C
Profit = 5C/2
The profit earned is 5C/2. The profit percent can be calculated as:
Profit Percent = (Profit / Cost Price) x 100
Profit Percent = (5C/2) / (20C) x 100
Profit Percent = 25%
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A landowner wishes to use 3 miles of fencing to enclose an isosceles triangular region of as large an area as possible. What should be the lengths of the sides of the triangle? Let x be the length of the base of the triangle. Write the area as a function of x. [First write the length of the equal-length sides in terms of the base, x, then write the height of the triangle in terms of the base.] V3 A(x) = 4 x Length of base = 1 miles Length of the other two (equal-length) sides = 2 x miles each
The area of the triangle, A(x), can be expressed as A(x) = (x * sqrt(3x^2/4))/2 and the height of the triangle, h, can be expressed as h = sqrt(3x^2/4) in terms of the base.
The landowner wishes to use 3 miles of fencing to enclose an isosceles triangular region of as large an area as possible. Let x be the length of the base of the triangle.The length of the base of the triangle is x miles, while the length of the other two equal-length sides are 2x miles each. Thus, the total length of the three sides of the triangle is 3x miles, which equals 3 miles of fencing as required.
To find the area of the triangle, we must first calculate the height of the triangle. Using the Pythagorean Theorem, we can calculate the height of the triangle in terms of the base. The formula is h^2 = (2x)^2 - (x/2)^2. Thus, the height of the triangle, h, can be expressed as h = sqrt(3x^2/4).The area of the triangle is equal to the base multiplied by the height and divided by two. Thus, the area of the triangle, A(x), can be expressed as A(x) = (x * sqrt(3x^2/4))/2.
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What is the answer I keep getting 32
Answer:
2 9/14
Step-by-step explanation:
Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order.
y dA, D is bounded by y = x − 6; x = y2
D
The value of the double integral using the easier order, ydA bounded by y = x − 6; x = y² is 125/12.
The double integral, indicated by ', is mostly used to calculate the surface area of a two-dimensional figure. By using double integration, we may quickly determine the area of a rectangular region. If we understand simple integration, we can easily tackle double integration difficulties. Hence, first and foremost, we will go over some fundamental integration guidelines.
Given, the double integral ∫∫yA and the region y = x-6 and x = y²
y = x-6
x = y²
y² = y +6
y² - y - 6 = 0
y² - 3y +2y - 6 = 0
(y-3) (y+2) = 0
y = 3 and y = -2
[tex]\int\int\limits_\triangle {y} \, dA\\ \\[/tex]
= [tex]\int\limits^3_2 {y(y+6-y^2)} \, dx \\\\\int\limits^3_2 {(y^2+6y-y^3)} \, dx \\\\(\frac{y^3}{3} + 3y^2-\frac{y^4}{4} )_-_2^3\\\\\frac{63}{4} -\frac{16}{3} \\\\\frac{125}{12}[/tex]
The value for the double integral is 125/12.
Integration is an important aspect of calculus, and there are many different forms of integrations, such as basic integration, double integration, and triple integration. We often utilise integral calculus to determine the area and volume on a very big scale that simple formulae or calculations cannot.
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hi could someone help me
The listed price of the TV that Wayne bought was $947.37
What was the listed price of the TV?Here we know that Wayne pays $660 for a TV whose price was marked down by (30 + 1/3)%
We can rewrite that discount as 0.30333... in a decimal form (get that just by dividing the percentage by 100%)
Then if the lited price of the TV is P, we can write the equation for the discount as follows:
660 = P*(1 - 0.3033...)
The number that we subtract is the percentage in decimal form.
Solving this for P gives:
P = 660/(1- 0.3033...) = 947.37
The listed price was 947.37 dollars.
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A certain medicine is given in an amount proportional to a patient's body weight. Suppose a patient weighing 162 pounds requires 216 milligrams of medicine. What is the weight of a patient who requires 220 milligrams of medicine?
A patient weighing 220 pounds needs 293 milligrams of medicine.
We have given that,
patient weighing 162
pounds requires 216 milligrams of medicine
We have to calculate the amount of medicine required by a patient weighing 220 pounds
Consider the value of amount of medicine is x.
Set up a proportion.
pounds / milligrams of medicine
What is the proportion we get?
[tex]162/216=220/x[/tex]
So,
[tex]162/216=220/x[/tex]
[tex]162x=216\times220[/tex]
[tex]162x=47,520[/tex]
[tex]x=47,520/162[/tex]
[tex]x=293[/tex]
A patient weighing 220 pounds needs 293 milligrams of medicine.
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Divide negative 6 and 2 over 5 ÷ negative 4 and 3 over 7. Negative 224 over 155 224 over 155 negative 519 over 35 519 over 35
By dividing we get: -6/2 ÷ 5 = -6/10 = -3/5, -4/3 ÷ 7 = -4/21, -224 ÷ 155 = -1.445, 224 ÷ 155 = 1.448, -519/35 = -14.83, 519/35 = 14.83.
.
To divide fractions, we need to remember the rule: "invert and multiply." This means that we take the reciprocal of the second fraction and then multiply it by the first fraction. For example, to divide -6/5 by 2/5, we invert 2/5 to get 5/2 and then multiply it by -6/5 to get -6/5 x 5/2 = -15/2. Similarly, to divide 224/155 by -1, we invert -1 to get -1/1 and then multiply it by 224/155 to get -224/155. To divide 519/35 by itself, we simply get 1 as the answer.
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Complete question:
Divide:
1. negative 6 and 2 over 5
2. negative 4 and 3 over 7
3. negative 224 over 155
4. 224 over 155
5. negative 519 over 35
6. 519 over 35
HELP!!
Write a quadratic equation in standard form that has solutions of -3 and -4.
Answer:
If a quadratic equation has solutions of -3 and -4, then it can be written in factored form as:
(x + 3)(x + 4) = 0
To convert this to standard form, we can multiply out the factors:
x^2 + 7x + 12 = 0
Therefore, the quadratic equation in standard form that has solutions of -3 and -4 is:
x^2 + 7x + 12 = 0
6(x - 2) -3 ≥ -4 ( -3 + 9) +10
Answer:
2x=5 > -14
Step-by-step explanation:
6(x - 2) -3 ≥ -4 ( -3 + 9) +10
6x-12-3 > 12-36+10
6x-15 > -36+22
6x-15 > -14
6x=15 > -14
3 3
2x=5 > -14
What is the surface area?
5 yd
6 yd
5 yd
5 yd
4 yd
square yards
The surface area of the rectangular prism is 170 square yards.
What is the surface area formula?Surface area is the total area of a three-dimensional shape's surface. Add the areas of all six faces to find the surface area of a cuboid with six rectangular faces. Alternatively, label the cuboid's length (l), width (w), and height (h) and use the formula: surface area (SA)=2lw+2lh+2hw.
To calculate the surface area of the rectangular prism, add the areas of each of its faces.
The front and back faces are 5 yards by 6 yards in size, so each has an area of:
5 yards x 6 yards equals 30 square yards
The top and bottom faces are 5 yards by 5 yards, so each has an area of:
5 yards x 5 yards equals 25 square yards
The two side faces have dimensions of 6 yards by 5 yards, for a total area of:
30 square yards = 6 yards x 5 yards
As a result, the surface area of the rectangular prism is as follows:
Front face area plus back face area plus top face area plus bottom face area plus left side face area plus right side face area
= 30+30+25+25+30+30
= 170 square yards
As a result, the rectangular prism has a surface area of 170 square yards.
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A+9 as a verbal expression
Answer:
"9 more than A" is a verbal expression.
Tell me pls what is the answer to this question? 3=x+3-5x??????????
Answer:
[tex]\boxed{x=0}[/tex]
Step-by-step explanation:
We need solve for x
[tex]3=x+3-5x[/tex]
substract 3 to both sides of the equation:
[tex]3-3=x+3-5x-3\\0=-4x[/tex]
divide both sides of the equation by -4
[tex]\frac{0}{-4}=\frac{-4x}{-4}\\0=x\\\equiv x=0[/tex]
The value of "x" that satisfies the equation is x=0,
[tex]\text{-B$\mathfrak{randon}$VN}[/tex]
Is the relation a function, and what is the range.
last one is the answer
Step-by-step explanation:
not a function because every input has more than 1 output
Snyder’s Moving company has two segments of customer: Small Business (SB) and Residential. 10% of Snyder’s 2,200 clients fall into both categories, using Snyder’s as their mover of choice for both their business and personal needs. The remaining clients are split evenly between segments. A client who falls into one category will use Snyder’s once every 3 years. Clients who fall into both categories use Snyder’s twice every 3 years. SB moves cost 15% more to the clients than Residential moves due to additional insurance costs for Snyder’s. The average profit Snyder’s makes per move, regardless of which type, is $260. Snyder recently launched a marketing campaign to all SB-only clients to encourage them to use Snyder’s for residential moves. 10% of recipients decided to do so. Assuming the campaign cost $500 to execute, how profitable was it?
The marketing campaign by Snyder's Moving company was profitable, generating an additional profit of $31,288 after subtracting the cost of the campaign. The total revenue generated after the campaign was $403,588.
To determine the profitability of the marketing campaign, we need to calculate the additional profit generated by the clients who were convinced to use Snyder's for residential moves.
Let's start by calculating the number of clients in each segment
Clients who fall into both categories, 0.1 x 2,200 = 220
Clients in each segment, (2,200 - 220) / 2 = 990
Now let's calculate the revenue generated by each segment
Revenue from clients in each segment: 990 clients x $260 profit per move = $257,400
Revenue from clients who fall into both categories: 220 clients x 2 moves x $260 profit per move = $114,400
Total revenue generated: $371,800
Now let's calculate the revenue generated after the marketing campaign
10% of 990 SB-only clients decided to use Snyder's for residential moves
0.1 x 990 = 99 clients
Additional revenue generated from these clients: 99 clients x $260 profit per move x 1.15 (15% higher price for SB moves) = $31,788
Total revenue generated after the marketing campaign, $403,588
Finally, let's subtract the cost of the marketing campaign
Profit generated after the marketing campaign: $403,588 - $500 = $403,088
Therefore, the marketing campaign was profitable, generating an additional profit of $31,288.
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12. INTERPRETING A LINEAR FUNCTION The table shows the length y
(in inches) of a person's hair after x months.
a. Write and graph a linear function that relates y to x.
b. Interpret the slope and the y-intercept.
Months, Hair
X
0
3
6
Length, y
11.0
12.5
14.0
Answer:
a. To write a linear function that relates the length of hair to the number of months, we need to find the slope and y-intercept of the line that passes through the given points. We can use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept.
Using the given data, we can find the slope as:
m = (y2 - y1) / (x2 - x1) = (14.0 - 11.0) / (6 - 0) = 0.5
And we can find the y-intercept as:
b = y - mx = 11.0 - 0.5(0) = 11.0
So, the linear function that relates the length of hair to the number of months is:
y = 0.5x + 11.0
We can also graph this function as a straight line, passing through the given points (0, 11.0), (3, 12.5) and (6, 14.0), as shown below:
linear function graph
b. The slope of the function is 0.5, which means that the length of hair increases by 0.5 inches every month. In other words, the slope represents the rate of change of hair length with respect to time.
The y-intercept of the function is 11.0, which means that if the person doesn't cut their hair at all (i.e., x = 0), their hair length would still be 11.0 inches. In other words, the y-intercept represents the starting value of the hair length.
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A coffee maker is on sale for 45$. If the sales tax is 7%, how much will the buyer spend altogether?
Answer: 38 I think if it's not right I'm sorry I'm bad at math that's like the only thing I suck at
Step-by-step explanation:
in how many ways can a class of 40 students select a committee from the class that consists of a president, a vice president, a treasurer and a secretary g
The total number of ways of selecting the committee is, therefore,40 x 39 x 38 x 37= 7,903,040
A class of 40 students select a committee from the class that consists of a president, a vice president, a treasurer, and a secretary in the following way:Step-by-step explanation:The number of ways that a class of 40 students can choose a committee consisting of a president, vice president, treasurer, and a secretary can be found by using the permutation formula.If we assume that the positions of the committee members are different, the number of ways can be calculated as follows:The number of ways of selecting the president from 40 students is 40.The number of ways of selecting the vice president from the remaining 39 students is 39.The number of ways of selecting the treasurer from the remaining 38 students is 38.The number of ways of selecting the secretary from the remaining 37 students is 37.The total number of ways of selecting the committee is, therefore,40 x 39 x 38 x 37= 7,903,040Thus, secretary.
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A triangle can have sides with measures: 10, 12, 24
True
False
Answer:
False.
Step-by-step explanation:
A triangle's two smaller sides have to add to be bigger than the larger side.
Answer:
False, that cannot exist by the hypotenuse rule
Hope it helps!
h(x)= -x + 5, solve for x when h(x) = 3
According to the given information, the solution to H(x) = 3 is x = 2.
What is equation?
In mathematics, an equation is a statement that asserts the equality of two expressions. An equation typically consists of two parts: the left-hand side (LHS) and the right-hand side (RHS). The LHS and RHS are separated by an equals sign (=), indicating that they have the same value. The general form of an equation is: LHS = RHS
To solve for x when H(x) = 3, we substitute 3 for H(x) in the equation and solve for x:
H(x) = -x + 5
3 = -x + 5
Subtracting 5 from both sides, we get:
-2 = -x
Multiplying both sides by -1, we get:
2 = x
Therefore, the solution to H(x) = 3 is x = 2.
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[tex]\huge\text{Hey there!}[/tex]
[tex]\mathtt{h(x) = -x + 5}\\\\\mathtt{3 = -x + 5}\\\\\mathtt{-x + 5 = 3}\\\\\textsf{SUBTRACT 5 to BOTH SIDES}\\\\\mathtt{-x + 5 - 5 = 3 - 5}\\\\\textsf{SIMPLIFY it}\\\\\mathtt{-x = 3 - 5}\\\\\mathtt{-x = -2}\\\\\mathtt{-1x = -2}\\\\\textsf{DIVIDE }\mathsf{-1}\textsf{ to BOTH SIDES}\\\\\mathtt{\dfrac{-1x}{-1} = \dfrac{-2}{-1}}\\\\\textsf{SIMPLIFY it}\\\\\mathtt{x = \dfrac{-2}{-1}}\\\\\mathtt{x = 2}[/tex]
[tex]\huge\text{Therefore your answer should be:}\\\\\huge\boxed{\mathtt{x = 2}}\huge\checkmark[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]
x + 3y = 12 x = -2y + 8
Answer:
x = 0
y = 4
Step-by-step explanation:
x + 3y = 12 x = -2y + 8
-2y + 8 + 3y = 12
y + 8 = 12
y = 4
Now put 4 in for y and solve for x
x + 3(4) = 12
x + 12 = 12
x = 0
Let's Check
0 + 3(4) = 12
12 = 12
So, x = 0 and y = 4 is the correct answer!
a car is traveling at a 3 of 57 miles per hour. What is the car's speed in miles per minute? how many miles will the car travel in 20 minute? (do not round your answer)
The speed of car is found as : 1 of 1140 miles per minutes.
The total number of miles the car will travel in 20 minutes is: 1 / 50 miles.
Explain about the unit conversions?The same attribute is expressed using a unit conversion, but in a diverse unit of measurement.
For e.g., time can be highlighted in minutes rather than hours, and distance can be verbalized in kilometers, feet, or another comparable measurement unit instead of miles.
Speed of car = 3 of 57 miles per hour.
Speed of car = 3 mi / 57 hr
We know, 1 hour = 60 minutes;
So,
Speed of car = 3 mi / 57*60 min
Speed of car = 1 mi / 57*20 min
Speed of car = 1 mi / 1140 min
Thus, the speed of car is found as : 1 of 1140 miles per minutes.
In 20 minute:
Number of miles = 1 mi / 1140 min * 20 min
Number of miles = 20 / 1140 min
Number of miles = 1 / 50 miles
Thus, the total number of miles the car will travel in 20 minutes is: 1 / 50 miles.
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A beverage company delivers three type of drinks which are Milk(M), Carbonateddrinks(C ) and Juice (J) to four stores ( A, B, C, D) for a period of two month. The number of pets of each type of beverage delivered to four stores in first monthis
represented in Matrix K and second month is Matrix L
8 4 3
2 2 2
1 3 1
3 1 3
M C J
A
B
K
C
D
1 3 1
2 7 2
8 3 4
9 2 6
MCJ
A
B
L
CD
i. Calculate the total number of pets delivered over the period of 2 months to eachstore. Suppose, the price charged for pet of each type of drink is given by the matrix. 225
195
212
M
C
J
ii. Calculate the cost of each store ( A, B, C, D) on the beverages in two months
The cost of each store in two months is: Store A is 4281, Store B is 5485, Store C is 3153, and Store D is 6413.
To calculate the total number of pets delivered over the period of 2 months to each store, we need to add the corresponding elements of matrix K and matrix L. The result will be a new matrix representing the total number of pets delivered to each store.
So, we have: Matrix K:
| 8 4 3 |
| 2 1 3 |
| 1 3 2 |
| 7 2 6 |
Matrix L:
| 1 3 2 |
| 8 4 9 |
| 2 6 3 |
| 4 9 2 |
Total pets delivered to each store:
| 9 7 5 |
| 10 5 12 |
| 3 9 5 |
| 11 11 8 |
To calculate the cost of each store in two months, we need to multiply the total number of pets delivered to each store by the price of each type of drink, as given by the matrix:
| 225 195 212 |
So, the cost of each store in two months is:
Store A: (9 × 225) + (7 × 195) + (5 × 212) = 4281
Store B: (10 × 225) + (5 × 195) + (12 × 212) = 5485
Store C: (3 × 225) + (9 × 195) + (5 × 212) = 3153
Store D: (11 × 225) + (11 × 195) + (8 × 212) = 6413
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How many students are in the band this year? how do you know?
Answer:
Step-by-step explanation:
which band?
At maxs party 3/12 of the guests are 10 year old and 7/12 of guests are 11 year old what fraction of the guests are 10 or 11 years old
The fraction of the guests are 10 or 11 years old is 5/6
A fraction represents a part of a whole, where the whole is divided into equal parts.
To find the fraction of guests who are 10 or 11 years old, we need to add the fractions of guests who are 10 years old and 11 years old.
3/12 of the guests are 10 years old, which can be simplified to 1/4.
7/12 of the guests are 11 years old.
Therefore, the fraction of guests who are 10 or 11 years old is
1/4 + 7/12 = 3/12 + 7/12
Add the fractions
= 10/12
Simplify the fraction
= 5/6
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