Let A = 0 1 1 0 (a) Calculate the matrix exponential e^(At). (Hint: It might...

Let A =

0	1
1	0

(a) Calculate the matrix exponential e^(At). (Hint: It might help to write down the power series expansions for the hyperbolic functions
cosh(t) =(e^t + e^(−t))/2
and sinh(t) =(e^t −e^(−t))/2
and then try to write eAt in terms of these two functions.)
(b) Use your matrix from part (a) to solve the nonhomogeneous initial value problem
x' =

0	1
1	0

x +

2

-1

, x(0) =

1

2

. (Hint: You might need the identity cosh^2(t)−sinh^2(t) = 1.)

Expert Solution

Colby Messinger answered 3 years ago

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere....

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere. Find the density function and cdf of a) X^1/2 b)X=e^x c)X=1/X Which of the random variables-X, X^1/2, e^x, 1/X does not have a finite mean?

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential?...

Let set E be defined as E={?∙?? +? | [?]∈R2}, where ?? is the natural exponential? function, please show E is a vector space by checking all the 10 axioms. (Notice: you may use the properties of vector addition and scalar multiplication in R2)

Let A be an n × n matrix which is not 0 but A2 = 0....

Let A be an n × n matrix which is not 0 but A2 = 0. Let I be the identity matrix. a)Show that A is not diagonalizable. b)Show that A is not invertible. c)Show that I-A is invertible and find its inverse.

Let 3x3 matrix A = -3 0 -4 0 5 0 &nb

Let 3x3 matrix A = -3 0 -4 0 5 0 -4 0 3 a) Find the eigenvalues of A and list their multiplicities. b) Find a basis, Bi, for each eigenspace, E(i). c) If possible, diagonalise matrix A. (i.e find matrices P and D such that Pinv AP = D is diagonal).

Let C be the following matrix: C=( 1 2 3 -2 0 1 1 -2 -1...

Let C be the following matrix: C=( 1 2 3 -2 0 1 1 -2 -1 3 2 -8 -1 -2 -3 2 ) Give a basis for the row space of Cin the format [1,2,3],[3,4,5], for example.

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0....

Q. Let A be a real n×n matrix. (a) Show that A =0 if AA^T =0. (b) Show that A is symmetric if and only if A^2= AA^T

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4...

Xn is a Markov Chain with state-space E = {0, 1, 2}, and transition matrix 0.4 0.2 0.4 P = 0.6 0.3 0.1 0.5 0.3 0.2 And initial probability vector a = [0.2, 0.3, 0.5] For the Markov Chain with state-space, initial vector, and transition matrix discuss how we would calculate the follow; explain in words how to calculate the question below. a) P(X1 = 0, X2 = 0, X3 = 1, X4 = 2|X0 = 2) b) P(X2 =...

Given a matrix A = [?1 ? ? 0 ?2 ? 0 0 ?2], with ?1...

Given a matrix A = [?1 ? ? 0 ?2 ? 0 0 ?2], with ?1 ≠ ?2 and ?1, ?2 ≠ 0, A) Find necessary and sufficient conditions on a, b, and c such that A is diagonalizable. B) Find a matrix, C, such that C-1 A C = D, where D is diagonal. C) Demonstrate this with ?1 = 2, ?2 = 5, and a, b, and c chosen by you, satisfying your criteria from A).

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A)...

Let A be a 2×2 symmetric matrix. Show that if det A > 0 and trace(A) > 0 then A is positive definite. (trace of a matrix is sum of all diagonal entires.)

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a...

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a so that f(x) is a probability density function b)What is P(X<=1)

Question