Determine whether A-3a is defective or not.

Answered: Determine whether A -3 a is defective…

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter30: Customary And Metric Steel Rules

Section: Chapter Questions

Problem 6A

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Question

Determine whether A
-3
a is defective or not.

Expert Solution

Step 1

Given:

$A = [\begin{matrix} - 1 & 1 \\ - 1 & - 3 \end{matrix}]$

To Determine:

A is defective or not.

Defective matrix:

A defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors.

Step 2

Here,

$A = [\begin{matrix} - 1 & 1 \\ - 1 & - 3 \end{matrix}]$

First, the find eigenvalues and eigenvectors

Start from forming a new matrix by subtracting $λ$ from the diagonal entries of the given matrix:

$A = [\begin{matrix} - λ - 1 & 1 \\ - 1 & - λ - 3 \end{matrix}]$

Find the determinant of the obtained matrix:

$|\begin{matrix} - λ - 1 & 1 \\ - 1 & - λ - 3 \end{matrix}| = λ^{2} + 4 λ + 4$

$λ^{2} + 4 λ + 4 = {(λ + 2)}^{2}$

$\Rightarrow {(λ + 2)}^{2} = 0$

Thus, the root are $λ1 = - 2 a n d λ2 = - 2$

These are the eigen values.

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Similar questions

Compute the complement and supplement of 61. a. Complement = -61, supplement = 119 b. Complement = -61, supplement = 299 c. Complement = 119, supplement = 29 d. Complement = 29, supplement = 119

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Determine whether A -3 a is defective or not.