Prove that a) If S is invertible and k is a positive integer, show that Sk is invertible and (Sk)−¹ : = (S-1)k b) Let S be an invertible matrix and let A, B be matrices such that B = S-¹AS. Show that Bk = S-¹ Ak S.

Q: P.) Consider the topology r = {GCR: 0 € G or G=0} on R. Is [0, 1) compact in (R, 7)? Explain why.

A: Given, topology  on defined by, Consider the open cover of  in (Since,  is open set in  for every…

Q: 1. Let f(z) = Z³ (z-i) Find Laurent Laurent series f(z) valid in the region A = { zec : 12-il > 1}

A: Given  The region is given by…

Q: #6: Use the Laplace transform to solve the following initial value problem, y"8y'-9y= S(t-4), y(0) =…

A:

Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: Let A, B and C be nonempty sets and let f : A → B, g: BC and h: B → C be functions. For each of the…

A:

Q: Consider the following initial value problem. 0 1 ≤1 12 1 ≤ 1 < 6 0 6 ≤ 1< 00 y' + 3y = y(0) = 10…

A: The Laplace transform of a function f(t) is defined as L{f(t)}=∫0∞f(t)e−stdt .Formulas: (1)…

Q: Let f: C be defined by f(z) M that 2²/2, if Z+0; if z=0 0. } Show the real and the Cauchy - Riemann…

A:

Q: Let fcilien be a collection of path components of a topological space X then c; =X. it

A: To prove that , we need to show two inclusions:1. : This follows directly from the definition of the…

Q: . Use the definition of derivative to compute the derivative of the following functions at x = 1: a.…

A:

Q: Determine the magnitude of the resultant force and its direction, measured counterclockwise from the…

A: We have given

Q: (a) Write the Taylor polynomial of degree 6 centered at 7 for cosx. (b) Use ≈ 3.14159 and the above…

A: (a)          Let…

Q: (10) t=3 mo A t=4 mo B t=1 mo (30) (20) (40) D F t=3 mo C E t-3 mo t=3 mo (50

A:

Q: Find the standard matrix for the linear transformation T: R² R² that reflects points about the line…

A: Linear Transformation T: R2  to  R2 that ''reflect'' points about the line x2=x1Claim : find…

Q: (a) Convert the point whose polar coordinates are (4, 57/4) to rectangular coordinates. (x, y) =…

A:

Q: per month, where t is time in months and S(t) is the number of computers sold each month. 2 3 S'(t)…

A:

Q: 4. Let the vector space P([-1,1]) have the inner product = f(t)g(t)dt, using 21

A:

Q: where I is the intensity of the sound and 6-10 watt per square meter. The decibel level, D, of sound…

A:

Q: weighing 32 pounds stretches a spring 2 feet. Determine the amplitude and period of motion if the…

A:

Q: Desmond plans to sell his homemade dog biscuits. He has baked 143,265 biscuits. He is planning to…

A: The divisibility rule for 5 is that a number is divisible by 5 if its unit digit is 0 or 5. In other…

Q: that the amounts of time (in minutes) required for assembling the frames, installing the wheels, and…

A: Given information:Time required for assembling frames, installing wheels, and decorating for…

Q: Use the limit comparison test to determine whether (a) Choose a series bn with terms of the form bn…

A: Given sqence is Here

Q: Suppose a subspace is spanned by the set S of vectors shown below. s={[G]-[-2.]} S Find a subset of…

A:

Q: Determine how and which of the functions above should be added, multiplied, divided or composed to…

A: f(x)=sinx,f(x)=logx and f(x)=2xdetermine  how and which of the functions should be added,…

Q: 2. The curve determined by the equation y2 = x²(x + 3) intersects itself, forming a loop. Find the…

A: 2. The curve determined by the equation  intersects itself, forming a loop.Find the area enclosed by…

Q: 2 east one of the answers above is NOT correct. e ite the given third order linear equation as an…

A:

Q: 3. In the following problems use the appropriate algebra and the appropriate formula to find the…

A:

Q: Prove that the dual of (P1) has a finite optimal valı

A:

Q: 2) Use Cauchy Integral Formula to evaluate integral z³ - 9e² z-2 where C is the counter |z| = 3 fo C…

A:

Q: 2. Solve the simultaneous congruences x = 4 (mod 5) and 2x = 3 (mod 7)

A:

Q: One way to estimate the average value of a function on an interval is to split that interval into a…

A: Matlab code:-function avg = approximate_averagef(f, a, b, N)    % Calculate the width of each…

Q: The number of bacteria in a culture is 1000 and this number increases 250% every two hours. Use a…

A: The number of bacteria in the culture initially is . According to the provided search results, the…

Q: Each character in a password is either a digit [0-9] or lowercase letter [az]. How many valid…

A: We shall count this using basic mathematics.

Q: Write x³ - 3x² - 10x + 24 ≤ 0 in factored form by using either synthetic or long division and then…

A:

Q: Prove that f(x) = x³ + x/2 is C¹-structurally stable.

A:

Q: Evaluate the following integral Lő (x + a)³ Where a = 0.49 and b = 2 Give your answer to the nearest…

A: We have to evaluate the integral

Q: Q(x₁, x2, x3) = x² + x² + x² + 4x₁x2 − 2x13 subject to x1+x2+x3 = 0 and -x₁+x2+x3 = 0, Q(x, y, z) =…

A: To determine the definiteness of the quadratic forms subject to the given constraints, we can use…

Q: Find f(t). f(t) = L 6s - 1 s²(s + 1)³) {

A:

Q: Yo V=O V=O Gap V O a.) Use the method of separation of variables to find V(x, y) inside the…

A: The potential function V(x,y) satisfies the Laplace equation,Vxx+Vyy=0,      0≤x≤b , 0≤y≤aFrom the…

Q: In a field plane PG(2, F), a line ax+by+cz = 0, not through the vertices of the triangle of…

A: To show that , , and are concurrent, we can use Ceva's theorem. Ceva's theorem states that in a…

Q: For each of the following pairs of functions f,g: NR, state with a brief justification whether f(x)…

A: The Definitions: if there exist positive constants c and k such thatThis means that g(x) is an…

Q: Determine if U = ? V -- H 5 4 CT is an eigenvector of the matrix A = 4 0 0 -18 5 24 0 12

A:

Q: Let a₁ [4] 9 āz DNE Is b a linear combination of a₁ and a2? and Ob is not a linaer combination of a₁…

A:

Q: A mass weighing 4 pounds is attached to a spring whose constant is 2 lb/ft. The medium offers a…

A:

Q: 1: Consider the following differential equation, 5xy"' + (2 + x) y + y = 0. Note that xo = 0 is a…

A: is a regular point.

Q: Give two sets of polar coordinates for each of the points A-F in the figure. E C IN LA Im JU 4 5-6 1…

A: A graph of the polar coordinates of points A, B, C, D,E, F are given.The aim is to find the polar…

Q: 3. Display an argument to show that the Gamma Function is defined for s> 0. = 6²°0 e-p₁-1α dt T(s) =…

A:

Q: The function f(x) = 10x ln(1 + x) is represented СО f(x) = Σ ca". n=0 Find the specified…

A: Let  the function is given f(x) =10 x ln(1 + x)as we know thatln(1+x)= x - (x ^ 2)/2 + (x ^ 3)/3 -…

Q: A plane has 43 rows of seats and 71 seats per row. What is the probability that you get a window…

A:

Q: Pick two random vectors in R³ (no zeros - few repeated numbers) and call them u and 7. Compute the…

A:

Q: In a large city, the road system is set up like a Cartesian plane, where streets are indicated by…

A: The baseball field is located 2 blocks west of street B and 28 blocks north of street A.