Suppose that for a 2 x 2 matrix A, Au = 30 for u=• How is the pair (3, ū) called for a matrix A?Suppose further that the only non-zero vectors x, for which Ar = rx for some r, mustbe multiples of v above.• What more can you now saw about the number r = 3?Suppose further that Aw-3u = i for w =Write down the general solution ofthe differential equationx'(t) = Ax

Given: Av→=3v→ for v→=-1-1. Consider the matrix A as abcd Av→=abcd-1-1=-a-b-c-d If a+b=c+d, then…

Answered: Suppose that for a 2 x 2 matrix A, Au =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that for a 2 x 2 matrix A, Au = 30 for u=
• How is the pair (3, ū) called for a matrix A?
Suppose further that the only non-zero vectors x, for which Ar = rx for some r, must
be multiples of v above.
• What more can you now saw about the number r = 3?
Suppose further that Aw-3u = i for w =
Write down the general solution of
the differential equation
x'(t) = Ax

Expert Solution

Solution:

Given: $A \vec{v} = 3 \vec{v} for \vec{v} = [\begin{matrix} - 1 \\ - 1 \end{matrix}]$ .

Consider the matrix A as $[\begin{matrix} a & b \\ c & d \end{matrix}]$

$\begin{array}{rcl} A \vec{v} & = & [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} - 1 \\ - 1 \end{matrix}] \\ = & [\begin{matrix} - a - b \\ - c - d \end{matrix}] \end{array}$

$\begin{array}{rcl} If \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} (a + b = c + d) \end{array} \begin{array}{rcl} , \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} then \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} (a + b) \end{array} \begin{array}{rcl} [\begin{matrix} - 1 \\ - 1 \end{matrix}] \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} or \end{array} \begin{array}{rcl} \end{array} \begin{array}{rcl} (c + d) \end{array} \begin{array}{rcl} [\begin{array}{l} - 1 \\ - 1 \end{array}] \end{array}$

Keep a+b = c+d =3, that gives $3 \begin{array}{rcl} [\begin{array}{l} - 1 \\ - 1 \end{array}] \end{array}$ .

Hence, the pair $(3, \vec{v})$ is called for a matrix A.

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