Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra With Applications
Found in: Page 441
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Solve the system dxdt=[-11-21]x with x(0)=[01]. Give the solution in real form. Sketch the solution.

The solution of the system isx(t)=[sin(t)sin(t)+cos(t)] and the graph is

See the step by step solution

Step by Step Solution

Step 1: Find the Eigen values of the matrix. 

Consider the equationdxdt=[1121]x with the initial valuex(0)=[01] .

Compare the equationsdxdt=[1121]x and dxdt=Axas follows.


Assume λis an Eigen value of the matrix[1121] implies |AλI|=0.

Substitute the values[1121] for Aand [1001]forI in the equation |AλI|=0as follows.


Simplify the equation|[1121]λ[1001]|=0 as follows.


Further, simplify the equation as follows.


Therefore, the Eigen values of Aareλ=±i .

Step 2: Determine the Eigen vector corresponding to the Eigen value λ=i.

Substitute the valuesi forλ in the equation |[1λ121λ]|=0as follows.


As Ei=ker[1i121i]=span[11+i], the values v+iwis defined as follows.


Therefore, the value ofS isS=[0111] .

Step 3: Determine the solution for dx→dt=[-11-21]x→.

The inverse of the matrix S=[0111]is defined as follows.


As x(t)=eptS[cos(qt)sin(qt)sin(qt)cos(qt)]S1x0, Substitute the value[0111] for S, [1110]for ,S1 [01]for ,x0

0 forp and1 for qin the equation x(t)=eptS[cos(qt)sin(qt)sin(qt)cos(qt)]S1x0as follows.


Further, simplify the equation as follows.


Therefore, the solution of the system isx(t)=[sin(t)sin(t)+cos(t)] .

Step 4: Sketch the solution. 

Asλ1,2=±i , draw the graph of the solution x(t)=[sin(t)sin(t)+cos(t)]as follows

Hence, the solution of the system dxdt=[1121]xwith the initial valuex(0)=[01] is x(t)=[sin(t)sin(t)+cos(t)]and the graph of the solution is an ellipse in counterclockwise direction.

Most popular questions for Math Textbooks


Want to see more solutions like these?

Sign up for free to discover our expert answers
Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.