Given the below matrix, determine the following. Show all work.a. The eigenvalues for the matrix.b. The eigenvectors for the matrix.c. If there are repeated eigenvalues, determine the algebraic multiplicity, geometricmultiplicity, and the defect for the repeated eigenvalue.[4A = 2L2-1 61 6-18]

The given matrix is A=4-162162-18. (a) The characteristic equation of the matrix A is given by…

Answered: Given the below matrix, determine the…

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

Given the below matrix, determine the following. Show all work.
a. The eigenvalues for the matrix.
b. The eigenvectors for the matrix.
c. If there are repeated eigenvalues, determine the algebraic multiplicity, geometric
multiplicity, and the defect for the repeated eigenvalue.
[4
A = 2
L2
-1 6
1 6
-1
8]

Expert Solution

Step 1

The given matrix is $A = [\begin{matrix} 4 & - 1 & 6 \\ 2 & 1 & 6 \\ 2 & - 1 & 8 \end{matrix}]$ .

(a)

The characteristic equation of the matrix A is given by $|A - λ I| = 0$ .

Therefore,

$\begin{array}{rcl} |\begin{matrix} 4 - λ & - 1 & 6 \\ 2 & 1 - λ & 6 \\ 2 & - 1 & 8 - λ \end{matrix}| & = & 0 \\ \Rightarrow & (4 - λ) [(1 - λ) (8 - λ) + 6] + 1 [2 (8 - λ) - 12] + 6 [(- 2) - 2 (1 - λ)] = 0 \\ \Rightarrow & (4 - λ) [(8 - λ - 8 λ + λ^{2}) + 6] + 1 [(16 - 2 λ) - 12] + 6 (- 2 - 2 + 2 λ) = 0 \\ \Rightarrow & (4 - λ) [(λ^{2} - 9 λ + 8) + 6] + (4 - 2 λ) + 6 (- 4 + 2 λ) = 0 \\ \Rightarrow & [(4 λ^{2} - 36 λ + 32 - λ^{3} + 9 λ^{2} - 8 λ) + 24 - 6 λ] + (4 - 2 λ) + (- 24 + 12 λ) = 0 \\ \Rightarrow & - λ^{3} + 13 λ^{2} - 50 λ + 56 + (4 - 2 λ) - 24 + 12 λ = 0 \\ \Rightarrow & - λ^{3} + 13 λ^{2} - 40 λ + 36 = 0 \end{array}$