2. (a) Find the values of a and b such that the eigenvalues of A
=...
2. (a) Find the values of a and b such that the eigenvalues of A
= |a 1 are 2 and -5. (b) Find the values of a, b and c such that
the eigenvalues of A = | 0 1 0 | 0 0 1 | a b c are 3, -2, and
5.
Find the values of λ (eigenvalues) for which the given problem
has a nontrivial solution. Also determine the corresponding
nontrivial solutions (eigenfunctions).
y''+2λy=0; 0<x<π, y(0)=0, y'(π)=0
(a) Let λ be a real number. Compute A − λI.
(b) Find the eigenvalues of A, that is, find the values of λ for
which the matrix A − λI is not invertible. (Hint: There should be
exactly 2. Label the larger one λ1 and the smaller λ2.)
(c) Compute the matrices A − λ1I and A − λ2I.
(d) Find the eigenspace associated with λ1, that is the set of
all solutions v = v1 v2 to (A...
Find the characteristic equation and the eigenvalues (and
corresponding eigenvectors) of the matrix. 2 −2 5 0 3 −2 0 −1 2 (a)
the characteristic equation (b) the eigenvalues (Enter your answers
from smallest to largest.) (λ1, λ2, λ3) = the corresponding
eigenvectors x1 = x2 = x3 =
Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are real. (Let
n represent an arbitrary positive number.)
y''+λy=
0,
y(0)= 0,
y'(π)= 0
a. Show that for all values of ? there is an infinite sequence
of positive eigenvalues of the problem
? ′′(?) + ??(?) = 0
? ?(0) + ? ′ (0) = 0, ?(1) = 0 (? = ?????)
b. Find eigenvalues of the problem if ? = 1.
1. For the following function ?(?) = (?^2−8?+16) / (?^2−4)
a. Find the critical values
b. Use the FIRST DERIVATIVE TEST to determine the intervals
where the function is INCREASING and DECREASING.
c. Find the RELATIVE EXTREMA of the function and state where
they occur.
d. Find the ABSOLUTE EXTREMA of the function on the interval
[−1, 1.75]