A 4x4 matrix a with real entries and a4 = i, what are the possible characteristic polynomials of a

Answers

Answer 1

If [tex]\mathbf A^4=\mathbf I[/tex], then the characteristic polynomial of [tex]\mathbf A^4[/tex] is

[tex]p_{\mathbf A^4}(\lambda)=\det(\lambda\mathbf I-\mathbf A^4)=\det((\lambda-1)\mathbf I)=(\lambda-1)^4[/tex]

which means [tex]\mathbf A^4[/tex] has eigenvalues [tex]\lambda=1[/tex].

We know that if [tex]\chi[/tex] is an eigenvalue of [tex]\mathbf X[/tex], then [tex]\chi^n[/tex] is an eigenvalues of [tex]\mathbf X^n[/tex].

So if [tex]\lambda=1[/tex] is the only eigenvalue of [tex]\mathbf A^4[/tex], we know that [tex]\pm\sqrt[4]\lambda=\pm1[/tex] are the only possible eigenvalues of [tex]\mathbf A[/tex]. We can construct five possible characteristic polynomials for [tex]\mathbf A[/tex] in that case.

[tex]p_{\mathbf A}(\lambda)=(\lambda-1)^4[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)(\lambda-1)^3[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^2(\lambda-1)^2[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^3(\lambda-1)[/tex]
[tex]p_{\mathbf A}(\lambda)=(\lambda+1)^4[/tex]

Related Questions

Solve 4/7=x+3/21 need help ASAP

Answers

4/7 = x+3/ 21

21 (4/7 = x+3)

3 *4=x+3

x= 9

What is cos (x)?

A) 48/58

B) 58/48

C) 26/58

D) 58/26

Answers

Remark
In order to solve for the Cos(x), you need to find the missing side. You can do that by finding the Sin(x) first and then the Cos of x or you can just use the Pythagorean theorem. I'm going to do the latter.

Formula
a^2 + b^2 = c^2

Givens
a = ???
b = 32
c = 58

Sub and solve
a^2 = ???
b^2 = 1024
c^2 = 3364

a^2 + 32^2 = 58^2
1024 + a^2 = 3364 Subtract 1024 from both sides.
a^2 = 3364 - 1024
a^2 = 2340 Take the square root of both sides.
sqrt(a^2) = sqrt(2340) Break a into its prime factors.
a = sqrt(2 * 2 * 5 * 3 * 3 * 13)

Rule
For every pair of equal factors, one can be brought outside the root sign and the other is discarded.
a = 2 * 3 * sqrt(5*13)
a = 6 sqrt(65)

The cos of an angle = opposite / hypotenuse.
Cos(x) = 6*sqrt(65) / 58 <<<<< This should be the answer
If you are offered choices, you should list them

Another choice would be 6*8.0623 / 58 = 0.8340 <<<<< Answer

Identify the radius of the circle whose equation is (x - 2)2 + (y - 8)2 = 16.

Answers

[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{}{ h},\stackrel{}{ k})\qquad \qquad radius=\stackrel{}{ r}\\\\ -------------------------------\\\\ (x-\stackrel{h}{2})^2+(y-\stackrel{k}{8})^2=16\implies (x-2)^2+(y-8)^2=\stackrel{r}{4^2}[/tex]

The radius is 4.

That equation is in center-radius form. When that’s the case, the radius is the square root of the number in the right of the equals sign. Square root of 16 is 4.

$10,000 is invested at two different rates. $2,500 is invested at 4%, and the rest is invested at 5%. How much interest is earned at the end of one year? Show all work for full credit.

Answers

So, what we have to do is split 10000
0.04(2500) + 0.05(7500)
Multiply:
100 + 375 = 475
$475 is made in one year

0.032 divided 4 = ___

Answers

0.008. Hope this helps in any way.

N english instructor asserted that students' test grades are directly proportional to the amount of time spent studying. lisa studies 6 hr for a particular test and gets a score of 79. at this rate, how many hours would she have had to study to get a score of 92?

Answers

You can simply use the cross method, so you will have
[tex]x = \frac{92 \times 6}{79} = 6.9[/tex]
which means she has to study approximately 7 hours to get score of 92.

Hope this helps.

The baseball team has sold tickets to a spaghetti dinner. forty-eight student tickets have been sold at $1.50 each. if the baseball team has made $149 total from the sale of all tickets, and adult tickets cost $3.50, how many adult tickets were sold?

Answers

Let the number of adults be x
number of students is 48
Amount collected from students=48*(1.5)=$72
Amount collected from adults=3.5*x=3.5x

Total amount collected will be:
72+3.5x=149
solving for x we have:
3.5x=149-72
3.5x=77
thus
x=77/3.5
x=22
thus the number of adults was x =22

Can someone help me solve this ??

Answers

The external angle at the intersection of two secants is half the difference of the intercepted arcs. That is

∠E = (1/2)(4x° -2x°) = x°

The given measure of arc AB tells you

... 4x° = 56°

... x° = 14°

The appropriate choice is ...

... B. 14°

Can someone explain this to me???

Answers

The question requires us to explain the different exponential graphs:
Exponential functions are given by:
f(x)=a(b)^x
when:
a is negative, the graph will be reflected along y-axis
b>1 the graph will be an exponential growth
b<1 the graph will be an exponential decay
1] a]
y=3(2)^x
y-intercept:
at y-intercept x=0 thus
y=3(2)^0
y=3(1)=3
thus y intercept is at y=3

c] since there is no asymptote

d] Domain:
All real numbers

e] Range:
[0,∞)

That all positive real numbers.

2]
a] f(x)=2(0.25)^x
a=2
b=0.25
b<1
thus the function is exponential decay

b] The horizontal axis is at y=0.

c] Domain:
The domain is all real numbers.

d] Range:
The range are all positive values.

e] y-intercept
f(x)=2(0.25)^x
at y-intercept , x=0 thus
y=2(0.25)^0
y=2(1)
y=2

3] y=-5(2)^x

a] This is a decay function since a=-5

b] y-intercept
f(x)=-5(2)^x
f(x)=-5(2)^0
f(x)=-5

C] the asymptote is y=0

d] Domain are all real numbers

e] Range
Range are all negative real numbers

Help quick please!!! Find the value of x

Answers

Hello!

First you have to find the missing angle in the triangle

The angles in a triangle add to 180°

180 - 97 - 30 = 53

The angle we just found and angle x form a straight line meaning they add to 180°

180 - 53 = 127

The answer is 127°

Hope this helps!

Hi!

X is part of a straight angle, which should add up to 180°.

So first we need to figure out what the other side of this straight angle is. Since the other side is inside the triangle, it should be easy.

All angles in any triangle always add up to exactly 180°. To figure out what the last angle is in the triangle, we just need to add the two existing angle measurements together and subtract the sum from 180.

97 + 30 = 127

180 - 127 = 53

So the last angle in the triangle is 53°.

Now, we can figure out the value of x. Since x is part of a 180° straight angle, and we know that half of said angle is 53°, we can find x by subtracting 53 from 180.

180 - 53 = 127

x = 127

Hope this helps :)

Compare the functions shown below:

f(x)

cosine graph with points at 0, negative 1 and pi over 2, 1 and pi, 3 and 3 pi over 2, 1 and 2 pi, negative 1
g(x)

x y
−6 −11
−5 −6
−4 −3
−3 −2
−2 −3
−1 −6
0 −11
h(x) = 2 cos x + 1

Which function has the greatest maximum y-value?

a. f(x)
b. g(x)
c. g(x) and h(x)
d. f(x) and h(x)

Answers

We can find the local maxima and minima of any -continous- function by first finding where the slope is 0, as at this point maxima or minima exist.

Given an arbitrary function '[tex]f(x)[/tex]' we find the point of slope 0 by taking its first derivative and equaling to 0 ('[tex] \frac{d}{dx}f(x)=0 [/tex]').

Lets, first, find the local extremes of the first function:
[tex]f(x)=cos(x)[/tex]
[tex] \frac{d}{dx} f(x)=\frac{d}{dx}cos(x)=-sin(x)=0[/tex]
[tex]x=sin^{-1} (0)=0[/tex]

So our first function has a maxima at '[tex]x=0[/tex]' or at '[tex]y=f(0)=cos(0)=1[/tex]'.

Now we get the extremes for the second function:
[tex]h(x)=2cos(x)+1[/tex]
[tex]\frac{d}{dx} h(x)=\frac{d}{dx}[2cos(x)+1]=-2sin(x)=0[/tex]
[tex]x=0[/tex]

So our second function has a maxima at '[tex]x=0[/tex]' or at '[tex]y=h(0)=2cos(0)+1=3[/tex]'.

Clearly, '[tex]h(0)=3\ \textgreater \ f(0)=1[/tex]', this means the second function '[tex]h(x)[/tex]' has the largest maxima -y value-.

Answer:

f(x) and h(x)

Step-by-step explanation:

f(x) and h(x) have the greatest maximum y-value because if you graph out all the equations you can determine by graph which ones have the greatest y-value. When you graph the equations you find that f(x) and h(x) both have a greatest y-value of 3! While g(x) has a greatest y-value of -2.

*Please note that it is looking for the greatest y-VALUE not the greatest y-INTERCEPT.

find the volume of the composite figure. round your answer to the nearest hundredth

Answers

611.08

Be sure to treat the two shapes as completely different figures, then add their totals together

The correct answer is 590.77mm3.

Given f(x)=x-7 and g(x)=x2 find g(f(-1))

Answers

[tex]f(x)=x-7\\\\f(-1)=-1-7=-8\\\\g(x)=x^2\\\\g(f(-1))=g(-8)=(-8)^2=64[/tex]

The cost of fertilizing a lawn is $0.25 per square foot. find the cost to fertilize the triangular lawn whose base is left parenthesis 14 plus startroot 17 endroot right parenthesis feet and altitude is startroot 68 endroot feet.

Answers

To Calculate the cost of The cost of fertilizing, we need to Calculate first the Area of the Triangle.

the Area of the Triangle = 1/2 x base x altitude = 1/2 x (14+√17) x 68 = 524.08 square foot

The cost of fertilizing = the cost of the square foot x the total area = 0.25x524.08 = 131 $

PLEASE HELP!! All I need to finish off my year! WILL BRAINLIEST AND VOTE!
1. How would you prove that circles are similar?
3.Find the center and the radius for the circle, given the equation:
x^2-14x+y^2+8y=16

Answers

Part 1)
we know that
All circles are similar. Figures can be proven similar if one, or more, similarity transformations (reflections, translations, rotations, dilations) can be found that map one figure onto another. In our attempt to prove all circles are similar, a translation and a scale factor (from a dilation) will be found to map one circle onto another

Part 2)
a) ∡BAD=128°-----------> by central angle

b) ∡BCD
The inscribed angle measures half of the arc it comprises.
so
∡BCD=128°/2----> 64°

c) ∡BED
The measure of the external angle is the semi difference of the arcs that it covers.
arc BCD=360°-128°=232°
∡BED=(1/2)*[232-128]-----> 52°

d) What is the relationship between BA and BE?
in the right triangle ABE
∡BEA=∡BED/2-----> 52°/2-----> 26°
tan (∡BEA)=BA/BE------> tan 26°=BA/BE----> BA/BE=0.4877

Part 3) Find the center and the radius for the circle, given the equation:
x²-14x+y²+8y=16

Group terms that contain the same variable

(x²-14x)+(y²+8y)=16

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

(x²-14x+49)+(y²+8y+16)=16+49+16

Rewrite as perfect squares

(x-7)²+(y+4)²=81

(x-7)²+(y+4)²=9²

a) the center is the point (7,-4)

b) the radius is 9

Which of the following steps could be used as a shortcut method to multiply 12 by 50

Answers

do 12 times 5 then add a zero to it

You can take the zero off 50 and do 12 times 5=60 and add the zero on and get =600

Write an equation for the line that passes through (0, -9) and (3, 0).

A. y = 3x + 9
B. y = -9x + 3
C. y = 3x + 3
D. y = 3x - 9

Answers

First find the slope of the line:-

= (0- - 9) / (3 - 0) = 9/3 = 3

Now use the point slope form of the equation to plug in the point (3,0):-
y - y1 = 3(x - x1)
y1 = 0 and x1 = 3:-
y - 0 = 3(x - 3)

y = 3x - 9 is the answer Its D.

In three more years, miguel's grandfather will be six times as old as miguel was last year. when miguel's present age is added to his grandfather's present age, the sum is 68. how old is each one now?

Answers

Grandfather is 57 and Miguel is 11 .Therefore, 57 + 11 =68 Miguel was 10 last year. In 3 years grandfather will be 60. 10 x 6= 60

Suppose you have an unfair coin that comes up heads 60% of the time and tails 40% of the time. you play a game where you flip the coin a single time and you lose $2 if you get heads and you win $3 if you get tails. what is the expected value of your outcome?

Answers

The expected value is that you lose 2$ because you come up with heads 60% of the time, which is the most likely outcome, and if you come up with heads then you lose 2$.

Use the verbs in the word bank to complete the sentences. Remember that you will need to put the verb in the correct form. Watch out for irregulars and stem changers! Do NOT write the sentence. ONLY fill in the verb in its correct form.

Word Bank:

ser comer jugar estar ir estudiar

1. Carlos _____ muy inteligente.

2. Mi hermana y yo ______ al museo.

3. Yo ________ al fútbol todos los sábados.

4. ¿Qué ______ tú cuando vas al restaurante mexicano?

5. Ustedes _________ los verbos en español. ¿No?

Question 1 options:
Blank # 1

Blank # 2

Blank # 3

Blank # 4

Blank # 5

Answers

1.es
2. fuimos
3. juego
4.comes
5. saben

*100% CORRECT ANSWER WERE

1) es

2) vamos

3) juego

4) comes

5) estudian

URGENT 50 POINTS!!!

Plan A has a $700 yearly premium with a $3,000 deductible.
Plan B has a $1,200 yearly premium with a $1,000 deductible.

Water damage (from broken pipes, overflows, etc.) results in approximately 12% of claims overall, with an average claim being $5,200. What is the expected value (loss or gain) to Acasa for every customer who chooses plan A?

A.)$-1,060
B.)$ 436
C.)$ -360
D.)$ -100

Answers

oh no my answer i tooooooooooo short ok bye
its not c.
answer- b

Answer:

The answer is B

Step-by-step explanation:

I got it right on plato

13. Which theorem or postulate proves the two triangles are similar? The diagram is not drawn to scale.
A. AA Postulate
B. AS Postulate
C. SSA Theorem
D. SSS Theorem

Answers

The Answer is the AA Postulate

Answer:

A) AA postulate

Step-by-step explanation:

This postulate used to prove the two triangles are similar.

If two angles in a triangle are congruent to two angles in other triangle, then the two triangles are called similar.

Hope this will helpful.

Thank you.

Tina is growing sunflowers in her garden. The flowers are currently 6 inches tall each week x, they will double in size . Which equation best represent how tall the sunflower will b in x weeks

Answers

For this case we have an equation of the form:
y = A (b) ^ x
Where,
A: initial height
b: growth rate
x: time
Substituting values we have:
y = 6 (2) ^ x
Answer:
An equation that best represent how tall the sunflower will b in x weeks is:
y = 6 (2) ^ x

Explain how you find a unit rate when given a rate.

Answers

This can be illustrated with an example:-

A car travels 100 miles in 2 hours
rate = 100 miles per 2 hours
unit rate = number of miles per ONE hour

so the unit rate in this case = 100/2 = 50 miles per hour.

Unit rate is the distance travelled in unit time. In this case the unit is an hour.

Kevin owes $4,645 on a credit card with a 19.7% interest rate compounded monthly. What is the monthly payment he should make in order to pay off this debt in 12 months, assuming he does not charge any more purchases with the card?

$387.08

$84.34

$429.62

$1,034.65

**Please help with steps too, please.

Answers

The "steps" are to enter the data into the appropriate places in a financial calculator. (PV=4645, i=19.7/12, n=12) Then PMT = -429.62.

Alternatively, and somewhat more slowly, you can make use of the amortization formula.
A = P(r/n)/(1 -(1 +r/n)^(-nt))
where A is the monthly payment,
P is the current balance (4645),
r is the annual rate (.197),
n is the number of compoundings per year (12), and
t is the number of years (1).

Filling in the numbers, you have
A = 4645*(.197/12)/(1 -(1 +.197/12)^-12)
A ≈ 429.62 . . . . . corresponds to the 3rd choice

Juan had $5.25 in nickels and quarters. if he had 15 more nickels than quarters, how many quarters did he have?

Answers

We will need to have a model for this problem to easily solve it.
First, let us say x is the number of nickels Juan had, and y is the number of quarters Juan had.

Now, we all know that a nickel is worth 5 cents and a quarter is worth 25 cents. The given said Juan had a total amount of nickels and quarters equal to $5.25 and that there should be 15 more nickels than quarters - we need a second model: $5.25 = $0.05x+$0.25y, but x = y+15 => $5.25 = $0.05(y+15)+$0.25y. Simplifying, we have $5.25 = $0.05y+$0.25y+$0.75 or $0.3y+$0.75.

Isolating the unknown, we have $5.25-$0.75 = $0.3y, or $4.50 = $0.3y.
Divide both sides by $0.3, we have y = 15. Therefore, x = 15+15 or 30.
Check the solution, 30($0.05)+15($0.25) = $5.25.

Therefore, Juan had 15 quarters.

Juan had 15 quarters. To solve this, we created a system of equations representing the total value of the coins and the relationship between the number of nickels and quarters. Calculations led to the conclusion that there were 15 quarters in total.

The question requires us to solve a system of equations to find out how many quarters Juan had if he had $5.25 in nickels and quarters, with 15 more nickels than quarters. Let's denote the number of quarters as q and the number of nickels as n. Remember, a quarter is worth $0.25 and a nickel is worth $0.05.

Based on the given information, we can establish the following equations:

0.25q + 0.05n = $5.25 (the total value equation)n = q + 15 (the number of coins equation)

We can substitute the second equation into the first to find the value of q.

0.25q + 0.05(q + 15) = $5.25

Solving for q:

0.25q + 0.05q + 0.75 = $5.25

0.30q + 0.75 = $5.25

0.30q = $4.50

q = $4.50 / 0.30

q = 15

Therefore, Juan had 15 quarters.

Consider the following set of equations:

Equation A: y = −x + 5
Equation B: y = 6x − 2

Which of the following is a step that can be used to find the solution to the set of equations?

−x = 6x + 2
−x − 2 = 6x + 5
−x + 5 = 6x – 2
−x + 5 = 5x

Answers

Since Y equals BOTH (-X+5) and (6X-2), you can set the two equations equal to eacher other; therefore, the answer is: C: -X+5=6X-2

The sides of a rectangle are in a 3:4 ratio. What is the length of the shorter sides of the rectangle if it's perimeter is 42cm

Answers

Final answer:

Given a rectangle with sides in a 3:4 ratio and a perimeter of 42 cm, the length of the shortest side is 9 cm.

Explanation:

We are given that the sides of the rectangle are in a 3:4 ratio and that the perimeter is 42 cm. The rectangular perimeter is calculated by the formula 2*(length + width). In this case, because the sides are in a 3:4 ratio, we can say 2*(3x + 4x) = 42, where 'x' represents the length of the shortest side.

Solve this equation, 2*(7x) = 42, we find that 'x' is 3. So the length of the shorter side, 3x, is 9 cm. Applying the same 3:4 ratio, the longer side, 4x, is 12 cm.

Learn more about Rectangle side lengths here:

https://brainly.com/question/16605856

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Final answer:

To find the length of the shorter side of a rectangle with sides in a 3:4 ratio and a perimeter of 42cm, we set up an equation using the ratio values as coefficients and the known perimeter. Solving this equation shows that a unit in the ratio equals 3cm and the shorter side is 9cm.

Explanation:

In this mathematics problem, we are trying to find the length of the shorter side of a rectangle, knowing that the sides are in a 3:4 ratio and that the perimeter is 42cm. The perimeter of a rectangle can be found by using the equation [tex]P=2L+2W[/tex] (Perimeter equals twice the length plus twice the width).

If we represent the shorter side as 3x and longer side as 4x, we can establish the following equation based on the given perimeter:[tex]2*(3x)+2*(4x)=42c[/tex]m. Simplifying this equation gives 14x=42cm. Solving for 'x', we find x=3. Therefore, each unit in the ratio corresponds to 3cm, and the length of the shorter side, represented by 3x, is [tex]3*3=9[/tex]cm.

Learn more about ratio here:

https://brainly.com/question/32531170

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All log graphs contain the points (1,0), (a,1), and (1/a,-1).

True .
False.

Answers

The correct answer is: True

Explanation:
1. [tex]log_a1 = 0[/tex] (Straightforward)
2. [tex]log_aa = 1 [/tex] (log to base of itself is always 1)
3. [tex]log_a \frac{1}{a} = log_aa^{-1} = -1(1) = -1[/tex] (Using log power property)

Hence the correct option is True.

A relay team ran a race. Each team member ran #1/4# of the distance. The whole race was #8/9# mile. How far did each team member run?

Answers

Each member ran 2/9 miles

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