This shows how much you would make per day for both choices.
Total earned in 31 days for $10,000 per day: $310,000
Explanation
$10,000 every day in 31 days.
Total earned in 31 days for penny: 21,474,836.47
Explanation
Day 1: $0.01
Day 2: $0.02
Day 3: $0.04
Day 4: $0.08
Day 5: $0.16
Day 6: $0.32
Day 7: $0.64
Day 8: $1.28
Day 9: $2.56
Day 10: $5.12
Day 11: $10.24
Day 12: $20.48
Day 13: $40.96
Day 14: $81.92
Day 15: $163.84
Day 16: $327.68
Day 17: $655.36
Day 18: $1,310.72
Day 19: $2,621.44
Day 20: $5,242.88
Day 21: $10,485.76
Day 22: $20,971.52
Day 23: $41,943.04
Day 24: $83,886.08
Day 25: $167,772.16
Day 26: $335,544.32
Day 27: $671,088.64
Day 28: $1,342,177.28
Day 29: $2,684,354.56
Day 30: $5,368,709.12
Day 31: $10,737,418.24
Let A be a given matrix below. First, find the eigenvalues and their corresponding eigenspaces for the following matrices. Then, find an invertible matrix P and a diagonal matrix such that A = PDPâ’1.
(a) [ 3 2 2 3 ]
(b) [ 1 â 1 2 â 1 ]
(c) [1 2 3 0 2 3 0 0 3]
(d) [3 1 1 1 3 1 1 1 3]
It looks like given matrices are supposed to be
[tex]\begin{array}{ccccccc}\begin{bmatrix}3&2\\2&3\end{bmatrix} & & \begin{bmatrix}1&-1\\2&-1\end{bmatrix} & & \begin{bmatrix}1&2&3\\0&2&3\\0&0&3\end{bmatrix} & & \begin{bmatrix}3&1&1\\1&3&1\\1&1&3\end{bmatrix}\end{array}[/tex]
You can find the eigenvalues of matrix A by solving for λ in the equation det(A - λI) = 0, where I is the identity matrix. We also have the following facts about eigenvalues:
• tr(A) = trace of A = sum of diagonal entries = sum of eigenvalues
• det(A) = determinant of A = product of eigenvalues
(a) The eigenvalues are λ₁ = 1 and λ₂ = 5, since
[tex]\mathrm{tr}\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3 + 3 = 6[/tex]
[tex]\det\begin{bmatrix}3&2\\2&3\end{bmatrix} = 3^2-2^2 = 5[/tex]
and
λ₁ + λ₂ = 6 ⇒ λ₁ λ₂ = λ₁ (6 - λ₁) = 5
⇒ 6 λ₁ - λ₁² = 5
⇒ λ₁² - 6 λ₁ + 5 = 0
⇒ (λ₁ - 5) (λ₁ - 1) = 0
⇒ λ₁ = 5 or λ₁ = 1
To find the corresponding eigenvectors, we solve for the vector v in Av = λv, or equivalently (A - λI) v = 0.
• For λ = 1, we have
[tex]\begin{bmatrix}3-1&2\\2&3-1\end{bmatrix}v = \begin{bmatrix}2&2\\2&2\end{bmatrix}v = 0[/tex]
With v = (v₁, v₂)ᵀ, this equation tells us that
2 v₁ + 2 v₂ = 0
so that if we choose v₁ = -1, then v₂ = 1. So Av = v for the eigenvector v = (-1, 1)ᵀ.
• For λ = 5, we would end up with
[tex]\begin{bmatrix}-2&2\\2&-2\end{bmatrix}v = 0[/tex]
and this tells us
-2 v₁ + 2 v₂ = 0
and it follows that v = (1, 1)ᵀ.
Then the decomposition of A into PDP⁻¹ is obtained with
[tex]P = \begin{bmatrix}-1 & 1 \\ 1 & 1\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}1 & 0 \\ 0 & 5\end{bmatrix}[/tex]
where the n-th column of P is the eigenvector associated with the eigenvalue in the n-th row/column of D.
(b) Consult part (a) for specific details. You would find that the eigenvalues are i and -i, as in i = √(-1). The corresponding eigenvectors are (1 + i, 2)ᵀ and (1 - i, 2)ᵀ, so that A = PDP⁻¹ if
[tex]P = \begin{bmatrix}1+i & 1-i\\2&2\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}i&0\\0&i\end{bmatrix}[/tex]
(c) For a 3×3 matrix, I'm not aware of any shortcuts like above, so we proceed as usual:
[tex]\det(A-\lambda I) = \det\begin{bmatrix}1-\lambda & 2 & 3 \\ 0 & 2-\lambda & 3 \\ 0 & 0 & 3-\lambda\end{bmatrix} = 0[/tex]
Since A - λI is upper-triangular, the determinant is exactly the product the entries on the diagonal:
det(A - λI) = (1 - λ) (2 - λ) (3 - λ) = 0
and it follows that the eigenvalues are λ₁ = 1, λ₂ = 2, and λ₃ = 3. Now solve for v = (v₁, v₂, v₃)ᵀ such that (A - λI) v = 0.
• For λ = 1,
[tex]\begin{bmatrix}0&2&3\\0&1&3\\0&0&2\end{bmatrix}v = 0[/tex]
tells us we can freely choose v₁ = 1, while the other components must be v₂ = v₃ = 0. Then v = (1, 0, 0)ᵀ.
• For λ = 2,
[tex]\begin{bmatrix}-1&2&3\\0&0&3\\0&0&1\end{bmatrix}v = 0[/tex]
tells us we need to fix v₃ = 0. Then -v₁ + 2 v₂ = 0, so we can choose, say, v₂ = 1 and v₁ = 2. Then v = (2, 1, 0)ᵀ.
• For λ = 3,
[tex]\begin{bmatrix}-2&2&3\\0&-1&3\\0&0&0\end{bmatrix}v = 0[/tex]
tells us if we choose v₃ = 1, then it follows that v₂ = 3 and v₁ = 9/2. To make things neater, let's scale these components by a factor of 2, so that v = (9, 6, 2)ᵀ.
Then we have A = PDP⁻¹ for
[tex]P = \begin{bmatrix}1&2&9\\0&1&6\\0&0&2\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}[/tex]
(d) Consult part (c) for all the details. Or, we can observe that λ₁ = 2 is an eigenvalue, since subtracting 2I from A gives a matrix of only 1s and det(A - 2I) = 0. Then using the eigen-facts,
• tr(A) = 3 + 3 + 3 = 9 = 2 + λ₂ + λ₃ ⇒ λ₂ + λ₃ = 7
• det(A) = 20 = 2 λ₂ λ₃ ⇒ λ₂ λ₃ = 10
and we find λ₂ = 2 and λ₃ = 5.
I'll omit the details for finding the eigenvector associated with λ = 5; I ended up with v = (1, 1, 1)ᵀ.
• For λ = 2,
[tex]\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}v = 0[/tex]
tells us that if we fix v₃ = 0, then v₁ + v₂ = 0, so that we can pick v₁ = 1 and v₂ = -1. So v = (1, -1, 0)ᵀ.
• For the repeated eigenvalue λ = 2, we find the generalized eigenvector such that (A - 2I)² v = 0.
[tex]\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}^2 v = \begin{bmatrix}3&3&3\\3&3&3\\3&3&3\end{bmatrix}v = 0[/tex]
This time we fix v₂ = 0, so that 3 v₁ + 3 v₃ = 0, and we can pick v₁ = 1 and v₃ = -1. So v = (1, 0, -1)ᵀ.
Then A = PDP⁻¹ if
[tex]P = \begin{bmatrix}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1\end{bmatrix}[/tex]
[tex]D = \begin{bmatrix}5&0&0\\0&2&0\\0&2&2\end{bmatrix}[/tex]
4. Find a polynomial function of degree 3 with the given zeros. Write your answer in the form:
Ax) = ax + bx² + cx+d
x= -2, x=-1, x=2
Answer:
I'm assuming the equation "Ax) = ax + bx² + cx+d" is really meant to read:
A(x) = ax^3 + bx² + cx+d
A(x) = x^3 + x^2 -4x -4
Step-by-step explanation:
(x+2)*(x+1)*(x-2) [This reflect the fact that x= -2, x=-1, x=2 all result in a zero]
Multiply this out to get A(x) = x^3 + x^2 -4x -4
Can someone please help I will give you 30 points and brainliest if the answer is right also no links please
Answer:
B: BC ≅EC
Step-by-step explanation:
You know that the two angles starting in C are congruent (they're opposite by vertex C).
Given that the B angle and the E angle are congruent, in order for the two triangles to be congruent by A(ngle)S(ide)A(ngle) you want the side inbetween to be congruent. That is BC and EC. Option B
im literally on my last braincell please help omg
an arithmetic sequence is given
-18, -13, -8, -3...
write an explicit formula for the nth term
Answer:
its going up by 5 every time
Step-by-step explanation:i think that would help i dont know if it was obvious
Find and intercept the slope of this line.
Answer:
The slope is -4. The amount of money left goes down by 4 with each comic book.
Step-by-step explanation:
The answer is D
help me graph this plzzzzzz mate
Answer:
Should look like this
Step-by-step explanation:
y-intercept = 2
slope = 1
TJ wants to sell his antique 1966 Mustang. He begins to set his price by looking at ads and finds these prices: $12,000, $9,500, $7,500, $14,000, and $12,000. What is the mean, median, mode, and range price?
Answer:
Range: 6,500
The difference between the largest and the smallest number.
Mode: 12,000
The number that appears most.
Mean: 11,000
Because its the average of all numbers
Median: 9,500
The middle number
If 3x + 2 > X + 8 then
Help please
Answer: x>3
Step-by-step explanation:
3x + 2 > X + 8
2x > + 8 - 2
2x > 6
x > 3
3x + 3 = 2x + 12 i need help for this answer
Answer:
X=9
Step-by-step explanation:
3x+3=2x+12
3x-2x=12-3
x=9
What is the final amount if 700 is decreased by 6% followed by a 7% increase?
Give your answer rounded to 2 DP.
Answer:
700 -6% =658 and 658-7%=611.94
What is the m 2x-3=5x-12
Answer:
15 is the answer
Step-by-step explanation:
If it was designed as an alternate angle, then it would 63 degrees
What is the slope? Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Step-by-step explanation:
Slope = (y2-y1)/(x2-x1)
Slope = (4-7)/(7-1)
Slope = -3/6
Slope = -1/2
Help please and thank you!
Answer:
... are ...Step-by-step explanation:
Verify if the ratio of corresponding sides is same:
Long sides ratio: 38/19 = 2Middle sides ratio: 28/14 = 2Short sides ratio: 20/10 = 2Since the ratios are same the triangles are similar
What are all the factors of 54?
1, 2, 3, 6, 9, 18, 27, 54
1, 2, 6, 9, 27, 54
1, 2, 6, 9, 26. 54
1, 2, 3, 6, 9, 18, 26, 54
Answer:
1,2,3,6,9,18,27,54
Step-by-step explanation:
1 x 54 = 54
2 x 27 = 54
3 x 18 = 54
6 x 9 = 54
9 x 6 = 54
18 x 3 = 54
27 x 2 = 54
54 x 1 = 54
Each container on a cargo ship holds 60 boxes of furniture.
Which table shows the values for 60c, the number of boxes
in c containers?
4
5
6
Number of Containers, C
Number of Boxes, 60c
240 300 360
4
5
6
Number of Containers, C
Number of Boxes, 60c
240 280 320
B.
4
5
6
Number of Containers, C
Number of Boxes, 60C
240 350 460
C
Answer:
imagine cheating on your star assessment but same tho the answer is c
Step-by-step explanation:
because i solved it on paper
Answer:
Step-by-step explanation:
C
What is the product?
Enter your answer as a fraction, in simplified form, in the box.
3/8*(-36)
Answer:
-27/2
Step-by-step explanation:
3/8 × -36
(3×-36)/8
-108/8
-27/2
may someone please help me solve the inequality
-4 (x - 2) + 5 < 9
Answer:
[tex]\huge\boxed{x > 1}[/tex]
Step-by-step explanation:
-4(x - 2) + 5 < 9
Subtract 5.
-4(x - 2) < 4
Divide by -4 (Flip the sign when dividing by a negative number).
x - 2 > -1
Add 2.
x > 1
Hope it helps :) and let me know if you want me to elaborate.
Answer:
x>1
Step-by-step explanation:
I had that same question hope that helped
pls help asap MIDSEGMENTS
Answer:
tttttttttttttttttttttttttttttttttttttttttttttt
Kylie borrowed a book from a library. The library charged a fixed rental for the book and a late fee for every day the book was overdue. The expression below shows the charges Kylie paid for the book when she returned it x days after the due date:
2 + 0.25x
What does the constant term of the expression represent? (1 point)
The late fee for the book
The fixed rental for the book
The number of books borrowed
The number of days the book was overdue
Answer:
it's "2+6556_789*4236+8765/987"
Step-by-step explanation:
Find the inverse (pre-calc)
Answer:
f^(-1)(x) = e^(x/3)
Step-by-step explanation:
1. Replace 'f(x)' by 'y:' y = 3ln x
2. Interchange x and y: x = 3ln y => x/3 = ln y
3. Solve for y:
e^(x/3) = y
4. Replace this 'y' with 'f^(-1)(x)'
Then f^(-1)(x) = e^(x/3)
[tex]\bf 2x-4=10[/tex]
Answer:
x = 7
Step-by-step explanation:
Add 4 on both sides to get 2x = 14, then divide 2 on both sides to get x = 7.
Hope this helps :)
Determine whether the statement is always, sometimes, or never true.
If △JKL ≅ △MNP, then m∠L = m∠N.
Complete the explanation of your reasoning.
The statement is always/ sometimes/ never true.
∠L corresponds to ∠P.
By [CPCTC/ Triangle Sum Theorem/ SAS Triangle Congruence Theorem], ∠L ≅∠P.
If ∠P ≅∠N, then ∠L [≅/≇ ] ∠N by [CPCTC/ Triangle Sum Theorem/ SAS Triangle Congruence Theorem], ∠L ≅∠P.
and m∠L [≅/≇ ] m∠N.
Please help this is on HMH I will give Brainliest if it's right! :)
Answer:
I am pretty sure the correct answer is never true
Step-by-step explanation:
This is because when,
Δ JKL ≅ Δ MNP
Then, ∠J ≅ ∠M
∠K ≅ ∠N
∠L ≅ ∠P
so ∠L ≠ ∠N
i hope this is right haha
Create a probability model for the outcome of performing 2 spins. What is the probability of NOT landing on yellow or red at all?
Using the concept of probability, the probability of not landing on yellow or red at all is 1/4.
Creating a square model of the two spins :
Red(R) ; Blue(B) ; Green(G) ; Yellow(Y)___ R ___ B ___ G ___ Y
R_ RR __ RB __ RG __ RY
B_ BR __ BB __ BG __ BY
G_ GR __GB __ GG __GY
Y_ YR __ YB __ YG __ YY
Sample space = 4² = 16Combinations without red and yellow :
{BB, BG, GB, GG} Number of selections without red or yellow = 4Recall :
Probability = required outcome / Total possible outcomesP(without Red or Yellow) = 4/16 = 1/4
Hence, the probability of not landing on yellow or red at all is 1/4.
Learn more : https://brainly.com/question/14605518
How do you find the Remainder when a polynomial in x is divided by a binomial of the form x-r?
Answer:
see explanation
Step-by-step explanation:
If a polynomial f(x) is divided by (x - r) then the remainder is f(r)
what is 1.70 times 8.5
the tens digit was 1. The ones digit was 6. The number was between 600 and 700 what was the number
Answer:
616
Step-by-step explanation: hope it helps :)
Answer:665,830 my answer
Step-by-step explanation:
so. what have you got so far?
How can a digit have no value?
Write an expression to represent:
Eight minus the quotient of two and a number xx
Answer:
8-(2/x)
Step-by-step explanation:
Help!?! I have no clue how to solve it :/
For a school fund raising project, Aiden and Jacob collected money in the ratio 3.5;6.8. Jacob collected $115.50 more than Aiden. Find the total amount of money they collected
Answer:
Step-by-step explanation:
PLEASE HELP ME I NEED THIS DONE RIGHT NOW
I know it says the answers are on the back but the answers aren’t there!!
