Use the factorization A PDP 1 to compute Ak, where k represents an arbitrary integer. {a 7(b-a)} = {0 b }
A^k =
Answers
If A = PDP⁻¹ = , Then \(A^k = (PDP^{-1})^{k}\)
= \(PDP^{-1} PDP^{-1} .... PDP^{-1}\) = \(PD^{k}P^{-1}\)
since P⁻¹ P = I, the identity matrix.
The term identity matrix is also widely used, but the term identity matrix is now standard. The term identity matrix is ambiguous, as it is also used for any unit of a matrix and ring of all n × n matrices.
In some fields, such as group theory or quantum mechanics, the identity matrix is sometimes shown in bold, or called "id" (short for identity). Less commonly, some math books use U or E for identity matrix, which stands for "identity matrix" and the German word Einheitsmatriks, respectively.
So for A = PDP⁻¹ is given by:
\(\left[\begin{array}{ccc}a&0&\\7(a-b)&b&\ \end{array}\right] = \left[\begin{array}{ccc}1&0&\\7&1&\\\end{array}\right] \left[\begin{array}{ccc}a&0&\\0&b&\ \end{array}\right] \left[\begin{array}{ccc}1&0&\\-7&1&\\\end{array}\right]\)
we have
\(\left[\begin{array}{ccc}a&0&\\7(a-b)&b&\ \end{array}\right] ^k = \left[\begin{array}{ccc}1&0&\\7&1&\\\end{array}\right] \left[\begin{array}{ccc}a&0&\\0&b&\ \end{array}\right] ^k \left[\begin{array}{ccc}1&0&\\-7&1&\\\end{array}\right]\)
= \(\left[\begin{array}{ccc}1&0&\\7&1&\\\end{array}\right] \left[\begin{array}{ccc}a^k&0&\\0&b^k&\ \end{array}\right] \left[\begin{array}{ccc}1&0&\\-7&1&\\\end{array}\right]\)
Then computing the matrix multiplication, we have:
\(\left[\begin{array}{ccc}1&0&\\7&1&\\\end{array}\right] \left[\begin{array}{ccc}a^k&0&\\0&b^k&\ \end{array}\right] \left[\begin{array}{ccc}1&0&\\-7&1&\\\end{array}\right] = \left[\begin{array}{ccc}a^k&0&\\7a^k&b^k&\\\end{array}\right] \left[\begin{array}{ccc}1&0&\\-7&1&\\\end{array}\right]\)
= \(\left[\begin{array}{ccc}a^k&0&\\7a^k-7b^k&b^k&\\\end{array}\right] = A^k\)
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Related Questions
What are the new vertices of quadrilateral KLMN if the quadrilateral is translated two units to the right and four units upward? A. K = (-2,0), L' = (1,0), M = (1,-3), N = (-2,-3) B. K = (-2,-2), L' = (1,-2), M = (1,-5), N = (-2,-5) C. K = (-0,0), L' = (3,0), M = (3,-1), N = (0,-1) D. K = (-2,2), L' = (1,2), M = (1,-1), N = (-2,-1)
Answers
K' ( -2, 2) , L' (1,2) , M' (1,-1) , N' (-2,-1)
Firstly, we needt to identify the coordinates of the vertices of the quadilateral
Afterwards, we then perform the translation
The translation simply means we add 2 to the x-value and add 4 to the y-value
So if we had coordinates of a vertex as (x,y); upon translation, we will have (x + 2, y+ 4)
Let us identify the vertices;
K'
K (-4,-2)
L (-1,-2)
M (-1,-5)
N (-4,-5)
After translation, we have;
K' ( -2, 2) , L' (1,2) , M' (1,-1) , N' (-2,-1)
pls help I will do anything
Answers
Answer:
32
Step-by-step explanation:
because 1 quart equals 4 cups so 8x4=32
Large values of r2 imply that the observations are more closely grouped about the _____.a. average value of the independent variablesb. average value of the dependent variablesc. least squares lined. origine. None of the above
Answers
One pound of grass seed covers about 400 square feet of field.
You would need.
pounds of grass seed to cover the field.
Answers
Answer:
I need grass seed to cover a circular Field 290 feet in diameter and a store sells 50 Pound bags of grass seed, 1 pound of grass seed covers about 400 square feet of field.
What is the slope of a line that is perpendicular to the line whose equation is
y= 4x – 3?
Answers
Answer:
-1/ 4
Step-by-step explanation:
y = 4x-3 has a slope of 4 because the equation is in slope intercept form
y = mx+b where the slope is m
Perpendicular lines have slopes that are negative reciprocals
-1/ 4 is the slope of a line that is perpendicular to y = 4x-3
At Thrasher Gaming Computers, Eli is the manager of the assembly department and has a team of employees that all build laptop computers. He made a graph representing the number of laptops built, y, based on the number of employee hours logged, x. The graph is a straight line that passes through the origin and the point (4, 15). Which of the following graphs and statements fit the situation? After 15 hours logged by employees, the department can build 4 laptops. After 4 hours logged by employees, the department can build 15 laptops.
I need this answered fast
Answers
(x, y) ——> (4, 15)
determine the magnitude of the force f the man at c must exert to prevent the pole from rotating, i.e., so the resultant moment about a of both forces is zero. true or false
Answers
The resultant moment about a both forces is zero. The statement is true.
A man B exerts a force on the rope, P = 30 lb
To find:
The magnitude of force, F
Consider a statement:
"The resultant moment about A of both forces is Zero."
Diagram attached at the end of the solution
\($ \theta=\tan ^{-1}\left(\frac{3}{4}\right) \\\)
\(& \theta=36.87^0\)
To prevent the pole from rotating, so that the resultant moment is about A = 0
We will apply the equilibrium Equation \($\sum \mathrm{M}_{\mathbf{A}}=0$\)
Here force \($P \times \sin (45)$\) and \($F \times \sin (\theta)$\) will not produce the moment at A.
Taking clockwise moment as a negative and anticlockwise moment as a positive.
\(& \mathrm{P} \times \cos (45) \times(18)-\mathrm{F} \times \cos (\theta) \times 12=0 \\\)
\(& 30 \times \cos (45) \times(18)=\mathrm{F} \times \cos (\theta) \times 12 \\\)
\($ \mathrm{~F}=\frac{30 \times \cos (45) \times(18)}{\cos (36.87) \times 12} \\\)
F = 39.77 lb
Based on the given parameters,
The magnitude of force, F = 39.77 lb
So there exists the force by considering the resultant moment about an of both forces as zero.
The statement is true.
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aya has 14 2/5 feet of chain. She wants to make pieces foot long math. How many can she make? b Solve the problem using decimals
Answers
Aya can make 14 mats of 1 foot long.
What is division?Division is one of the fundamental arithmetic operation, which is performed to get equal parts of any number given, or finding how many equal parts can be made. It is represented by the symbol "÷" or sometimes "/"
Given that, Aya has 14\(\frac{2}{5}\) feet of chain. She wants to make pieces foot long mat.
Let can make x mats out of the given chain, since each mat is 1 foot long, so,
1×x = 14\(\frac{2}{5}\)
x = 72/5
x = 14.4
x ≈ 14
Hence, She can make 14 mats out of the given chain.
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I got the second part but I forgot how to get the third part
Answers
Answer:
y = 250x + 1500
Step-by-step explanation:
y = mx + b
y = 3000
m = 250
x = 6
3000 = 250(6) + b
Multiply.
3000 = 1500 + b
Subtract.
1500 = b
what is the HCF of 210 and 308
Answers
The least common multiple of 210 and 308 is 4620.
Given Triangle ABC with the coordinates shown, what is the length of the median from vertex A? Round your answer to the nearest hundredth.
Answers
The length of the median from vertex A is 13.03 units for the given triangle.
What is Midpoint?A midpoint is defined as in the middle of the line connecting two points a position known as a midpoint.
The coordinates of the vertex for the given triangle as:
A(2,8),
B(-4,-6),
C(6,-4)
Let m be the midpoint of BC.
So the coordinates midpoint m is:
(-4 + 6)/2, (-6+(-4))/2
(2/2, -10/2)
(1, -5)
Now, the length of the median from vertex A is:
√(2-1)² + (8-(-5))²
√(1)² + (13)²
√1+169
√170
13.03 (Rounded to the nearest hundredth)
Thus, the length of the median from vertex A is 13.03 units
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frances receives one dollar for every pound of worms she gives her grandfather. which reinforcement schedule is this? group of answer choices fixed interval fixed ratio variable interval variable ratio
Answers
Answer:fixed ratio
Step-by-step explanation: :)
This is an illustration of a fixed ratio reinforcement schedule: after a predetermined number of responses, Frances is given a reward (in this case, $1). (every pound of worms).
What is a fixed ratio reinforcement schedule?An example of a reinforcement plan utilized in the behavioral psychology concept of operant conditioning is a fixed ratio reinforcement schedule. A reward or reinforcement is delivered according to this schedule after a predetermined number of actions or responses. As an illustration, a rat might receive a food pellet after every 10 lever presses.
Fixed ratio schedules work well for encouraging high response rates since the subject is driven to keep up the activity in order to earn the reward. Yet, if the reinforcement is stopped, the behavior can become less frequent or cease entirely.
Fixed ratio schedules can also cause behaviors to become rigid or stereotyped because the subject may start concentrating only on the number of responses required to earn the reward and stop considering other options or actions.
Overall, fixed ratio reinforcement regimens can be helpful in teaching both humans and animals to carry out particular tasks, but they should be utilized with caution and consideration for any unexpected consequences.
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Help me pls... Thnx!
Answers
Given:
First term of a geometric series = a
Common ratio of the geometric series = \(\sqrt{3}\)
To prove:
\(S_{10}=121a(\sqrt{3}+1)\)
Solution:
The sum of first n terms of geometric series is
\(S_n=\dfrac{a(r^n-1)}{r-1}\)
Where, a is the first term and r is the common ratio.
Putting n=10 and \(r=\sqrt{3}\) in the above formula, to get the sum of 10 terms.
\(S_{10}=\dfrac{a((\sqrt{3})^{10}-1)}{\sqrt{3}-1}\)
\(S_{10}=\dfrac{a(243-1)}{\sqrt{3}-1}\)
\(S_{10}=\dfrac{242a}{\sqrt{3}-1}\)
Rationalizing the denominator, we get
\(S_{10}=\dfrac{242a}{\sqrt{3}-1}\times \dfrac{\sqrt{3}+1}{\sqrt{3}+1}\)
\(S_{10}=\dfrac{242a(\sqrt{3}+1)}{(\sqrt{3})^2-1^2}\)
\(S_{10}=\dfrac{242a(\sqrt{3}+1)}{3-1}\)
\(S_{10}=\dfrac{242a(\sqrt{3}+1)}{2}\)
\(S_{10}=121a(\sqrt{3}+1)\)
Hence proved.
The height of lava fountains spewed from volcanoes cannot be measured directly. Instead, their height in meters can be found using the equation y=1/2 gt^2
where y represents the height, g is 9.8, and t represents the falling time of the lava rocks. Find the height in meters of a lava rock that falls for 5 seconds.
Answers
The height in meters of a lava rock that falls for 5 seconds is; 122.5 m.
How to calculate height from quadratic equations?
We are told that the height is modelled by the equation;
y = ¹/₂gt²
where;
y represents the height.
g is 9.8 m/s²
t represents the falling time of the lava rocks.
Thus, at t = 5, we have;
y(5) = ¹/₂(9.8)(5)²
y(5) = 122.5 m
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the dotplot above shows the distribution of calf measurements for 507 adults who attend a local gym. the data is divided into equal-sized subgroups with 25% of the adults in each subgroup. which measure of center can you most accurately estimate from this information?
Answers
which measure of center can you most accurately estimate from this information: Median
The median divides the data into equal-sized groups. So approximately 50% of the data is above the median and approximately 50% of the data is below the median.
Calculating the median
The median is the value in a data set, implying that half of data focuses have a value more modest or equivalent to the median and half of data focuses have a value higher or equivalent to the median.
For a little data set, you first count the quantity of data focuses (n) and organize the data focuses in expanding request. Assuming the quantity of data focuses is lopsided, you add 1 to the quantity of places and separation the outcomes by 2 to get the rank of the data point whose value is the median.
The rank is the place of the data point after the data set has been organized in expanding request: the smallest value is rank 1, the second-smallest value is rank 2, and so forth.
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the complete question is:
The dotplot above shows the distribution of calf measurements for 507 adults who attend a local gym. The data is divided into equal-sized subgroups with 25% of the adults in each subgroup.
Which measure of center can you most accurately estimate from this information?
1. mean
2. median
The quotient of the absolute
value of negative six squared and
three
Answers
Answer:
-6 times -6= 36 divide by 3 =12 so the answer would be 12
2 Consider the given function g(x)=x² + 4x + 3 (e) Sketch the function. (f) Determine the axis of symmetry. (9) Determine the minimum or maximum value of the function. (h) Write the domain and range in interval notation. Write your answers in exact form.
Answers
The function g(x) = x² + 4x + 3 is a quadratic function. To sketch the function, we can plot some points on the graph and connect them to form a curve. The axis of symmetry is a vertical line that passes through the vertex of the parabola.
The minimum or maximum value of the function can be determined by finding the vertex. The domain and range of the function can be expressed in interval notation.
To sketch the function g(x) = x² + 4x + 3, we can calculate the y-values for different x-values and plot the points on a graph. This will give us an idea of the shape of the graph, which is a parabola that opens upwards.
The axis of symmetry can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic function. In this case, a = 1 and b = 4, so the axis of symmetry is x = -4/2 = -2.
The minimum or maximum value of the function occurs at the vertex of the parabola. The vertex can be found by evaluating x = -b/2a. In this case, the vertex occurs at x = -4/2 = -2. Substituting this value into the function, we get g(-2) = (-2)² + 4(-2) + 3 = -1. So, the minimum or maximum value of the function is -1.
The domain of the function is all real numbers because there are no restrictions on the x-values. The range of the function is the set of all real numbers greater than or equal to the minimum value, which is -1. Therefore, the domain is (-∞, ∞) and the range is [-1, ∞) in interval notation.
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Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If f(x)=2x−x2+1/3x3.⋯………. converges for all x, then f′′′(0)=2.
Answers
The statement given is false. The reason for this is that the convergence of a function does not necessarily imply anything about the value of its derivative. To disprove the statement, we can consider the function f(x) = x^2, which converges for all x, but its third derivative f'''(x) = 0, which means that f'''(0) is also equal to 0. Hence, f′′′(0) is not equal to 2.
In general, it is important to note that the convergence of a function does not provide any information about the behavior of its derivatives. Moreover, a function may converge at some points and diverge at others, and this can be determined by analyzing the behavior of its terms or by using convergence tests. In this case, it is necessary to compute f′′′(0) directly using the definition of the derivative or by applying differentiation rules.
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I need help with this question please
Answers
Answer:
the answer is C
Step-by-step explanation:
A rainwater collection system collects water into a holding tank. For every 1 cm of rainfall on a 100 m² roof, 1,000 L of water can be colleA. How many liters of water can be crainfall on a 120 m² roof?
Part B. How many centimeters of rainfall are needed to fill a 40,000 L holding tank from a 120 m² roof?
Answers
Answer:
A: 3 0 0
B: 150 ;-;
Step-by-step explanation:
What is 6 divided by 2
Answers
To check:
2 * 3 = 6
Answer:
3
Step-by-step explanation:3x2=6
Find the volume (to the nearest cubic foot) of a cone with base radius 4 ft. and height 8 ft. a. 43 cu. ft. c. 134 cu. ft. b. 128 cu. ft. d. 402 cu. ft.
Answers
Volume is a three-dimensional scalar quantity. The volume of a cone with a base radius of 4 ft. and height of 8 ft. is 134 ft³.
What is volume?A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.
The volume of a cone with a base radius of 4 ft. and height of 8 ft. can be written as,
Volume = (π/3)×(Radius)²×Height
= (π/3)×(4)²×8
= 134.0412 ft³ ≈ 134 ft³
Hence, the volume of a cone with a base radius of 4 ft. and height of 8 ft. is 134 ft³.
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Is negative 1 a real number?
Answers
1. A plane flies due east for 500 km and then on a heading of 120" for 150 km. What are its distance and bearing from its starting point?
2. A ship leaves port and sails due west for 120 km, then due south for 40 km. What are the distance and bearing of the port from the ship?
Answers
(1) The distance from the starting point to the final point is 589.5 km and the bearing from the starting point to the final point is 16.7° north of east
(2) The distance from the ship to the port is 126.49 km and the bearing of the port from the ship is 18.4° south of west.
What is the distance and bearing of the plane?
To solve this problem, we can use the law of cosines to find the distance from the starting point to the final point. We can also use trigonometry to find the bearing (angle) between the starting point and final point.
See the diagram attached:
We want to find the distance and bearing from the starting point (x) to the final point.
Using the law of cosines, we have:
c² = a² + b² - 2abcos(C)
c² = 500² + 150² - 2x500x150xcos(120°)
c² = 250,000 + 22,500 + 75,000
c² = 347,500
c = √347,500
c = 589.5 km
To find the bearing, we can use trigonometry. Let θ be the angle between the line from the starting point to the final point and due east.
tan(θ) = (150 km) / (500 km)
θ = tan⁻¹(0.3)
θ = 16.7°
2. To solve this problem, we can use the Pythagorean theorem to find the distance from the ship to the port. We can also use trigonometry to find the bearing (angle) between the ship and the port.
We want to find the distance and bearing of the port (P) from the ship (S).
Using the Pythagorean theorem, we have:
d² = 120² + 40²
d² = 14400 + 1600
d² = 16000
d = √(16000)
d = 126.49 km
To find the bearing, we can use trigonometry. Let θ be the angle between the line from the ship to the port and due west.
tan(θ) = (40 km) / (120 km)
θ = tan⁻¹ (1/3)
θ = 18.4°
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The left and right page numbers of an open book are two consecutive integers whose sum is 331 Find these page numbers.
Answers
Answer:
165 and 166
Step-by-step explanation:
You can divide 331 by 2 to get 165.5. You then round down for one answer, 165, and round up for the other, 166. This evens out the rounding, and to check your answer you can add 165 and 166, which is 331.
hi can you please help me with this. the question is in the picture:)
Answers
Solve for m.
13m – 34m + 42m - 22m + 8 = -41
Answers
Answer:
m=14
Correct me if I'm wrong ^•^
Answer:
m = -49
Explanation:
Combine like terms
13m - 34m + 42m - 22m + 8 = -41
-m + 8 = -41
Subtract 8 from both sides of the equation
-m + 8 = -41
-m + 8-8 = -41 -8
Simplify
Subtract the numbers
-m = -49
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Match each given function with the description of its graph.
Answers
The functions and their matching properties are:
y = (1/4)ˣ: y-intercept at (0,1); graph approaches 0 as x increasesy = 2(1/2)ˣ: y-intercept at (0,2); graph approaches 0 as x increasesy = 3ˣ: y-intercept at (0,1); graph approaches positive infinity as x increasesHow to match the functions with their properties?The functions are exponential functions;
Exponential functions are represented using:
y = abˣ
Where a represents the y-intercept.
Also, when b is less than 1, the graph would approach 0 as the x value increases, otherwise it approaches positive infinity.
This means that the graphs of y = (1/4)ˣ and y = 2(1/2)ˣ would approach 0 as the x value increases
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3y - 15 =y + 1 solve for y
Answers
A n s w e r:
y = 8
Hope this helps!
Question 2 of 10
James wants to tile his floor using tiles in the shape of a trapezoid. To make
the pattern a little more interesting he has decided to cut the tiles in half
along the median. The top base of each tile is 13 inches in length and the
bottom base is 19 inches. How long of a cut will John need to make so that
he cuts the tiles along the median?
OA. 32 inches
OB. 3 inches
O C. 16 inches
OD. 6 inches
SUBMIT
Answers
John needs to make a 16 inches cut of the tiles along the median. The correct answer is option C. 16 inches.
When cutting the tile along the median, we need to find the length of the cut that divides the trapezoid into two equal areas.
The median of a trapezoid is the line segment connecting the midpoints of the two non-parallel sides. In this case, the top base of the trapezoid is 13 inches and the bottom base is 19 inches.
To find the length of the cut, we can take the average of the lengths of the top and bottom bases. The average of 13 inches and 19 inches is (13 + 19) / 2 = 32 / 2 = 16 inches.
Therefore, John will need to make a 16-inch cut along the median to cut the tiles in half and create the desired pattern on his floor.
Option C, 16 inches, correctly represents the length of the cut required to cut the tiles along the median.
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