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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q41E (page 346)

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?
$A = [\begin{matrix} 1 & a \\ 0 & 1 \end{matrix}]$

Short Answer

Expert verified

The matrix a=0 is diagonalizable.

Step by step solution

01

Characteristic equation

Consider an n 脳n matrix A and a scalar 位. Then 位 is an eigenvalue5 of A.

$\det (A - 位濒) = 0$

This is called the characteristic equation (or the secular equation) of matrix A.

02

Solution of the problem

We solve:

$\det (A - 位濒) = 0 |\begin{matrix} 1 - 位 & a \\ 0 & 1 - 位 \end{matrix}| = 0 {(1 - 位)}^{2} = 0 位_{1, 2} = 1$

03

Substitute

For , $位 = 1$ we solve:

$(A - l) x = 0 [\begin{matrix} 0 & a \\ 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}] {ax}_{2} = 0 For a 鈮� 0, we have E_{1} = span ([\begin{matrix} 1 \\ 0 \end{matrix}])$

So A is not diagonalizable. For a = 0, though, we get A - l = 0, thus $E_{1} = {鈩�}^{2}$ , and therefore A is diagonalizable.

Here, the final result is a = 0

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Most popular questions from this chapter

Consider the matrix $A = | \begin{matrix} a & b \\ b & c \end{matrix} |$ where a, b, and c are nonzero constants. For which values of a, b, and c does A have two distinct eigenvalues?

Find a basis of the linear space Vof all $3 脳 3$ matrices Afor which both $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] and [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$ are eigenvectors, and thus determine the dimension of.

27: a. Based on your answers in Exercises 24 and 25, find closed formulas for the components of the dynamical system

$\overset{炉}{x} (t + 1) = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}] \overset{炉}{x} (t)$

with initial value $\overset{鈫�}{x_{0}} = \overset{鈫�}{e_{1}}$ . Then do the same for the initial value $\overset{鈫�}{x_{0}} = \overset{鈫�}{e_{2}}$ . Sketch the two trajectories.

b. Consider the matrix

$A = [\begin{matrix} 0.5 & 0.25 \\ 0.5 & 0.75 \end{matrix}]$

.

Using technology, compute some powers of the matrix A, say, A², A⁵, A¹⁰, . . . .What do you observe? Diagonalize matrix Ato prove your conjecture. (Do not use Theorem 2.3.11, which we have not proven

yet.)

c. If $A = [\begin{matrix} a & b \\ c & d \end{matrix}]$

is an arbitrary positive transition matrix, what can you say about the powers A^tas t goes to infinity? Your result proves Theorem 2.3.11c for the special case of a positive transition matrix of size 2 脳 2.

Consider the matrix $A = | \begin{matrix} a & b \\ b & a \end{matrix} |$ where aand bare arbitrary constants. Find all eigenvalues of A.

If $\overset{鈬赌}{v}$ is an eigenvector of matrix A with associated eigenvalue 3 , show that $\overset{鈬赌}{v}$ is an image of matrix A .

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