Answered: 7 -8 8 -9 -1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

determine all eigenvalues and corresponding eigenvectors of the given matrix.

Expert Solution

Step 1

Eigenvalue: A number $λ$ is said to be an eigenvalue of a square matrix A if there exists a non-zero

vector x such that $A x = λ x$ .

Here, the non-zero vector x is called the eigenvector corresponding to the eigenvalue $λ$ . We have

to find the eigenvalues of the matrix $A = [\begin{matrix} 7 & - 8 & 6 \\ 8 & - 9 & 6 \\ 0 & 0 & - 1 \end{matrix}]$ .

Step 2

Consider the matrix $A - λ I = [\begin{matrix} 7 - λ & - 8 & 6 \\ 8 & - 9 - λ & 6 \\ 0 & 0 & - 1 - λ \end{matrix}]$ . The determinant of this matrix is $- {(λ + 1)}^{3}$ . Solving

the equation $- {(λ + 1)}^{3} = 0$ , we get, $λ_{1} = - 1, λ_{2} = - 1, λ_{3} = - 1$ . These are the eigenvalues of A.

For $λ = - 1$ , we have, $A - λ I = [\begin{matrix} 8 & - 8 & 6 \\ 8 & - 8 & 6 \\ 0 & 0 & 0 \end{matrix}]$ . We have to find the null space of A. For this, we will solve the

matrix equation $[\begin{matrix} 8 & - 8 & 6 \\ 8 & - 8 & 6 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ . This gives $[\begin{matrix} 1 & - 1 & \frac{3}{4} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ . If we take $x_{2} = t, x_{3} = s$

then, we have $x_{1} = t - \frac{3}{4} s$ . Thus, $x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] \Rightarrow x = [\begin{matrix} t - \frac{3 s}{4} \\ t \\ s \end{matrix}]$ .

Step by step

Solved in 3 steps

Knowledge Booster

$Matrix Eigenvalues and Eigenvectors$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions