1. (15 pts) Is the matrix A = 1 0 1 0 1 1 1 1 2
diagonalizable? If yes, find an invertible matrix P and a diagonal
matrix Λ such that P −1AP = Λ.
Q5 [15 pts]
a) Convert the following NFA to a DFA:
0 1
----------------------
-> a || {a} | {a,b}
b || {c} | {c}
c || {d} | {d}
d || {e} | {e}
* e || {} | {}
b) Informally describe the language that it accepts.
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).
Matrix:
Ax b
[2 1 0 0 0 | 100]
[1 1 -1 0 -1 | 0]
[-1 0 1 0 1 | 50]
[0 -1 0 1 1 | 120]
[0 1 1 -1 1 | 0]
Problem 5
Compute the solution to the original system of equations by
transforming y into x, i.e., compute x = inv(U)y.
Solution:
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I have not Idea how to do this. Please HELP!
(15 pts) Suppose that the continuous random variable X has
pdf
?(?) = {
?; 0 < ? < 2 2?; 5 < ? < 10 0; otherwise
a) Determine the value of c that makes this a legitimate pdf.
b) Sketch a graph of this pdf.
c) Determine the cumulative distribution function (cdf) of X.
d) Sketch a graph of this cdf.
e) Calculate ? = ?(?) and ? = ??(?).
f) What is ?(? = ?)?
g) Compute...
eigenvalues of the matrix A = [1 3 0, 3 ?2 ?1, 0 ?1 1] are 1, ?4
and 3. express the equation of the surface x^2 ? 2y^2 + z^2 + 6xy ?
2yz = 16. How should i determine the order of the coefficient in
the form X^2/A+Y^2/B+Z^2/C=1?
Consider Matrix A = ([5, 0, 4],[1, -1, 0],[1, 1, 0]). Note that
[5, 0, 4] is row 1. [1, -1, 0] is row 2. [1, 1, 0] is row 3.
a) Find all Eigenvalues and Eigenvectors.
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...