The positive integral powers of a square matrix A are defined as follows; A^1=A, A^2=AA, A^3=AA^2, A^4= AA^3,..., A^n+1=A^n. suppose that r and s are positive integers. Prove that A^r A^s=A^(r+s) and that (A^r)^s = A^rs ( in close analogy with the laws of exponents for real numbers).
Answers
To prove that A^r A^s = A^(r+s), we can use mathematical induction.
Base case: r = 1, s = 1
A^1 A^1 = A^(1+1)
A A = A^2
This is true since A^2 is defined as AA.
Inductive step: assume A^r A^s = A^(r+s) is true for some positive integers r and s.
We want to prove that A^(r+1) A^(s+1) = A^((r+1)+(s+1)) = A^(r+s+2)
A^(r+1) A^(s+1) = A^r A A^s A
(using the definition of A^(r+1) and A^(s+1))
= A^r A^s A A
(using the inductive assumption A^r A^s = A^(r+s))
= A^(r+s) A A
(using the definition of A^(r+s))
= A^(r+s+1) A
(using the definition of A^(n+1) = A^n A)
= A^(r+s+2)
(using the definition of A^(n+1) = A^n A)
Therefore, A^r A^s = A^(r+s) for all positive integers r and s.
To prove (A^r)^s = A^(rs), we can also use mathematical induction.
Base case: r = 1
(A^1)^s = A^s
This is true since (A^1) is just A, and A^s is defined as A multiplied by itself s times.
Inductive step: assume (A^r)^s = A^(rs) is true for some positive integer r.
We want to prove that (A^(r+1))^s = A^((r+1)s)
(A^(r+1))^s = (A^r A)^s
(using the definition of A^(r+1) = A^r A)
= (A^r)^s A^s
(using the distributive property of matrix multiplication)
= A^(rs) A^s
(using the inductive assumption (A^r)^s = A^(rs))
= A^(rs+s)
(using the first part of the proof A^r A^s = A^(r+s))
= A^((r+1)s)
Therefore, (A^r)^s = A^(rs) for all positive integers r and s.
In conclusion, we have proven that A^r A^s = A^(r+s) and (A^r)^s = A^(rs) in close analogy with the laws of exponents for real numbers.
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Related Questions
a chart that compares three set of values in a three-dimensional chart is _____.
Answers
A chart that compares three sets of values in a three-dimension chart is called surface.
A two-dimensional collection of points (flat surface), a three-dimensional collection of points with a curved cross section (curved surface), or the perimeter of any three-dimensional solid are all examples of surfaces in geometry.
A surface is typically a continuous boundary that separates two areas of a three-dimensional space. For instance, a sphere's surface divides its interior from its exterior, and a horizontal plane divides the half-planes above and below it. Despite the fact that regions they contain are three-dimensional and have a volume, surfaces are basically two-dimensional and have an area. Despite this, surfaces are frequently referred to by the names of the regions they surround. Differential geometry studies the characteristics of surfaces, particularly the concept of curvature.
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Regression analysis has been used to calculate the line of best fit from a series of data. Using this line to predict a value which lies between the two extreme values
observed historically is known as extrapolation.
Is this statement TRUE or FALSE?
Answers
The statement is TRUE. Extrapolation in regression analysis refers to using the line of best fit to predict values that lie outside the range of the observed data.
It involves extending the regression line beyond the observed data points to estimate values for variables that are beyond the range of the data. However, it's important to note that extrapolation can be risky because it assumes that the relationship between the variables continues outside the observed range, which may not always be valid. Extrapolation should be done with caution and its results should be interpreted carefully.
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A cylinder has a base radius of 2 meters and a height of 2 meters. What is its volume
in cubic meters, to the nearest tenths place?
Answers
the answer is 25.1 meters3
What is the cardinality of the set {2, {3}}?
Answers
Given,
The set is {2, {3}}.
The cardinity of the set is the number of elements in the set.
Consider,
A = {2, {3} }
The cardinity of the set is,
\(|A|=2\)Hence, the cardinality of the set is 2.
ASAP Answer right with PROOF, Will give thanks, stars and brainiest
Answers
Answer:
5
Step-by-step explanation:
(67/8 -29/6)+35/24
=201-116+35/24
=120/24
=5
Write it in interval notation
x ≤ 3 and x ≠ -1 or 4 ≤ x
Answers
The interval notation that expresses the expression is (-∝, 3] ∩ (-∝, -1) ∩ (-1, ∝) ∪ [4, ∝)
How to use interval notation to represent the expressions?The expression is given as:
x ≤ 3 and x ≠ -1 or 4 ≤ x
In inequality, "and" is represented with ∩, while "or" is represented with ∪
The above can be represented as:
x ≤ 3 ∩ x ≠ -1 ∪ 4 ≤ x
As an interval notation, we have (-∝, 3] ∩ (-∝, -1) ∩ (-1, ∝) ∪ [4, ∝)
Hence, the interval notation that expresses the expression is (-∝, 3] ∩ (-∝, -1) ∩ (-1, ∝) ∪ [4, ∝)
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Suppose that, in a probability experiment, there are three independent events A, B and C. Furthermore, P(A) = 0.2, P(B) = 0.24, and P(C) = 0.32.
What is the probability that all three events will happen?
Answers
The probability that all three events will happen is 0.01536.
Let A, B and C be the three independent events, such that
P(A) = 0.2
P(B) = 0.24
P(C) = 0.32
The probability that all three events will happen can be calculated using the formula:
P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
Now, substituting the values, we get:
P(A ∩ B ∩ C) = 0.2 × 0.24 × 0.32
= 0.01536
Therefore, the probability that all three events will happen is 0.01536.
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Ai Mi has a part-time job at an ice skating rink selling hot cocoa. She decided to plot the number of hot cocoas she sold relative to the day's high temperature and then draw the line of best fit. Based on the line of best fit, how many hot cocoas would you predict Ai Mi to sell if the day’s high temperature were 64
∘
∘
F?
Answers
The number of hot cocoas the Ai Mi would be expected to sell with a high temperature of 64 ºF is given as follows:
60.
How to define the linear function?As the graph of the linear function is given, the most practical way to define the function is using the slope-intercept format, as follows:
y = mx + b.
In which the coefficients are given as follows:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.From the graph, when x = 0, y = 111, hence the intercept b is given as follows:
b = 111.
When x increases by 5, from 0 to 5, y decays by four, from 111 to 107, hence the slope m is obtained as follows:
m = -4/5
m = -0.8.
Hence the function that predicts the amount of hot cocoa sold for a high temperature of x ºF is given as follows:
y = -0.8x + 111.
The prediction for a high temperature of x = 64 is given as follows:
y = -0.8(64) + 111
y = 60 hot cocoas, rounded.
Missing InformationThe graph is given by the image presented at the end of the answer.
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PLEASE HELP!!
just look at the picture
Answers
Answer:
its the math that solve
Step-by-step explanation:
?
cuantos es 2X = 8?
Answers
Answer:
I love algebra anyways
The ans is in the picture with the steps how i got it
(hope this helps can i plz have brainlist :D hehe)
Step-by-step explanation:
help me with this question please
Answers
Answer:
N+1
Step-by-step explanation:
It begins with = 2
then n+1=3
then n+2=4 and so on it is n+1
Plz help i only need c im confused lol 6th grade history
Answers
Answer: Upper: Kings, priests, warriors, and government officials.
Middle: artisans, merchants, farmers, and fishers
Lowest: enslaved people, who worked on farms.
Step-by-step explanation:
C. prisoners of farmers, and rich people/they were maids and cleaned up everything.
solve for (2x-1)dx+(7y+3)dy=0
Answers
In summary This is the solution to the differential equation (2x-1)dx+(7y+3)dy=0.
Why is it?
To solve for (2x-1)dx+(7y+3)dy=0, we need to find a function whose differential is (2x-1)dx+(7y+3)dy.
We can do this by integrating both sides with respect to their respective variables:
∫(2x-1)dx + ∫(7y+3)dy = 0
Integrating the first term with respect to x gives:
x²2 - x + C1
where C1 is the constant of integration.
Integrating the second term with respect to y gives:
7/2 y²2 + 3y + C2
where C2 is the constant of integration.
So the general solution to the differential equation is:
x²2 - x + C1 + 7/2 y²2 + 3y + C2 = 0
We can simplify this by combining the constants into a single constant, say C:
x²2 - x + 7/2 y²2 + 3y + C = 0
This is the solution to the differential equation (2x-1)dx+(7y+3)dy=0.
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The melting point of ice is 0 degree celsius. during a chemistry experiment, jill observed ice melting within 2 degrees celsius of this measurement. find the range of temperature that jill observed. define a variable, write an inequality, and solve.
Answers
The typical definition of the melting point is the temperature at which a substance transforms from a solid to a liquid.
What do you mean by melting point?
melting point, temperature at which the solid and liquid forms of a pure substance can exist in equilibrium. As heat is applied to a solid, its temperature will increase until the melting point is reached.
The melting point of a liquid is the temperature at which the liquid transforms from a solid to a liquid under atmospheric pressure. This is the location where the liquid and solid phases are equally present. The substance's melting point varies with pressure as well and is reported at standard pressure.
As the melting point is 0 °C.
So, it melts at 2 C. It can melt from 0 °C – 2 °C to 0 °C + 2 °C.
Range = [-2, 2] in °C.
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if $a(-3, 5)$, $b(7, 12)$, $c(5, 3)$ and $d$ are the four vertices of parallelogram $abcd$, what are the coordinates of point $d$?
Answers
The coordinates of point D in the parallelogram ABCD are (15, 10).
To find the coordinates of point D, we can use the properties of a parallelogram. In a parallelogram, opposite sides are parallel and congruent. Therefore, we can use this information to determine the coordinates of point D.
Let's consider the given points:
A(-3, 5)
B(7, 12)
C(5, 3)
Since opposite sides of a parallelogram are parallel, the vector connecting points A and B should be equal to the vector connecting points C and D. We can express this as:
AB = CD
To find the vector AB, we subtract the coordinates of point A from the coordinates of point B:
AB = (7 - (-3), 12 - 5)
= (10, 7)
Now, we can express the vector CD using the coordinates of point C and the vector AB:
CD = (5, 3) + (10, 7)
= (15, 10)
Therefore, the coordinates of point D are (15, 10).
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A
5
B
10
C
6
D
4
pls help
Answers
Answer:
Pretty sure it's C.
Step-by-step explanation:
A and B are 6 units away from each other.
Answer:
C
Step-by-step explanation:
Count the boxes
assessment started: unit 8 progress check: mcq part a. item 1 let f be the function given by f(x)=3xsinx. what is the average value of f on the closed interval 1≤x≤7 ?
Answers
To find the exact value, we would need to evaluate the definite integral, but it may not be practical to do so without further information or using numerical methods.
To find the average value of a function on a closed interval, you need to calculate the definite integral of the function over that interval and divide it by the length of the interval. In this case, we want to find the average value of the function f(x) = 3xsin(x) on the interval 1 ≤ x ≤ 7.
The average value of f on the interval [1, 7] is given by the formula:
Average value = (1/(b - a)) * ∫[a to b] f(x) dx
where a and b are the endpoints of the interval.
In our case, a = 1, b = 7, and f(x) = 3xsin(x). So, we can calculate the average value as follows:
Average value = (1/(7 - 1)) * ∫[1 to 7] (3xsin(x)) dx
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Find the least common denominator between the two fractions. 1/2 and 5/6 2 6 8 12
Answers
Answer:6
Step-by-step explanation:
You have three $5 bills and four $10 bills in your wallet. You randomly select different dollar bills from your wallet. Find the probability that you select a $5 bull second given that you randomly selected a $10 first.
Answers
The probability to select $5 bill when the first selected bill is $10 is given by 1/2.
Probability is the possibility to occur of an event.
In mathematical terms, probability can be defined as,
\(\text{Probability}=\frac{\text{The number of favorable case}}{\text{The total number of case}}\)
Given that, the number of total $5 bills = 3
the number of $10 bills = 4
Total number = 3+4 = 7
If the first drawn bill is $10, then the number of $10 bills = 4-1 = 3
Total number of bills =7-1 = 6
So the probability to select $5 bill when the first selected bill is $10 = 3/6 = 1/2
Hence, the probability to select $5 bill when the first selected bill is $10 is given by 1/2.
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find the value of x that would prove y/z State the converse used to support your answer.
Answers
Answer:
x=6
alternative exterior angles
Step-by-step explanation:
to prove y and z parallel then the two angles must be equal:
11x-3=4x+39
11x-4x=39+3
7x=42
x=42/7
x=6
Im actually going to explode I hate geometry with a passion
Answers
Answer: B 28√15
Step-by-step explanation:
They want you to find the Area of the quadrilateral. But if you can find the area of 1 trangle then you can double it because the 2 triangles are congruent.
We know the 2 triangles are congruent from the theorem (see diagram
We also know that QR=TR=SR They are all radius
Let's solve for triangle PRS
PR = hypotenuse = 10+7 =17
SR=7 radius
Use pythagorean to find PS
c²=a²+b²
17²=7²+b²
289 = 49 + b²
b²=289 -49
b² = 240
b=√240
b=\(\sqrt{16*15}\)
b= 4√15 This is PS
Area of triangle = 1/2 bh b=PS h=7
Area of triangle = 1/2 (4√15)(7)
Area of triangle = (2√15)(7)
Area of triangle = 14√15
Area of quadrilateral = 2 (14√15) > 2 triangles make the quadrilateral
Area of quadrilateral = 28√15
Can someone plz help ☹️☹️☹️
Answers
Answer:
35
Step-by-step explanation:
x=3x-70
because corresponding angles are congruent
Solve the equation to find x.
how many integers between 1 and 1,000,000 have the sum of the digits equal to 15
Answers
There are 13,992 integers between 1 and 1,000,000 with a sum of digits equal to 15.
To find the number of integers between 1 and 1,000,000 with a sum of digits equal to 15, we can use a combinatorial approach.
We need to distribute the sum of 15 among the digits of the number. We can think of this as placing 15 indistinguishable balls into 6 distinct boxes (corresponding to the digits).
Using the stars and bars method, the number of ways to distribute the sum of 15 among 6 boxes is given by the combination formula:
C(n + r - 1, r - 1)
where n is the sum (15) and r is the number of boxes (6).
Plugging in the values, we have:
C(15 + 6 - 1, 6 - 1) = C(20, 5)
Calculating this combination, we find:
C(20, 5) = 13,992
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in each diagram line p is parallel ANSWER FAST Pls
Answers
Answer:
choice A is correct
Step-by-step explanation:
the value of x should be 75
PLEASE HELP WILL GIVE BRAINLIEST!!!
Answers
Answer: Choice B) Investment decreases by 3% each month
=====================================================
Explanation:
We won't use the value 2500 at all.
The base of the exponential is 0.97
Set this equal to 1+r and solve for r
1+r = 0.97
r = 0.97-1
r = -0.03
The negative r value indicates we have exponential decay, so the value of the investment decreases by 3% each month.
Let's say you had $100 to start off. This would mean the investment would be worth $97 after one month is up.
Is this the correct answer? Please someone help!!!
Answers
Answer:
yes
Step-by-step explanation:
same slope
Answer:
yes same slope
Step-by-step explanation:
NEED HELPPP #2 PLZZZZZZZ
Answers
Answer:
Step-by-step explanation:
p > 9
Hope that helps!
round to the nearest place value shown
Answers
Prove that for any complex polynomial ^=−1,
the sum of the roots is zero. (Please state any theorem used).
Answers
Answer:
I dont get this ??
Step-by-step explanation:
If you invest $900 in a bank where it will earn 8 percent compounded annually, how much will it be worth at the end of seven years? Complete the steps below using cell references to given data or previous calculations. In some cases, a simple cell reference is all you need. To copy/paste a formula across a row or down a column, an absolute cell reference or a mixed cell reference may be preferred. If a specific Excel function is to be used, the directions will specify the use of that function. Do not type in numerical data into a cell or function. Instead, make a reference to the cell in which the data is found. Make your computations only in the green cells highlighted below. In all cases, unless otherwise directed, use the earliest appearance of the data in your formulas, usually the Given Data section. Given Data: Annual Interest Rate 8% Number of years 7 Money available for investing S900.00 Value of investment after 7 years
Answers
The investment will be worth approximately $1,546.45 at the end of 7 years. To calculate the value of the investment after 7 years, we can use the formula for compound interest:
Value = Principal * (1 + interest rate)^time
Given Data:
Principal (P) = $900
Annual Interest Rate (r) = 8% or 0.08
Number of years (t) = 7
Substituting the values into the formula, we have:
Value = $900 * (1 + 0.08)^7
Calculating the exponent:
(1 + 0.08)^7 = 1.08^7 ≈ 1.718279
Now we can calculate the value of the investment:
Value = $900 * 1.718279 ≈ $1,546.45
Therefore, the investment will be worth approximately $1,546.45 at the end of 7 years.
In this calculation, we used the compound interest formula, which takes into account the initial principal, the annual interest rate, and the number of compounding periods (in this case, 7 years). The interest is compounded annually, meaning that at the end of each year, the interest earned is added to the principal for the next year's calculation.
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