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Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter7: Exponents And Exponential Functions

Section7.1: Multiplication Properties Of Exponents

Problem 1BGP

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Question

Find an orthogonal or unitary diagonalizing matrix for each of the following:

Expert Solution

Step 1

Suppose, the matrix is:

$A = [\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}]$ .

Firstly, solve the characteristic equation and find the eigenvalues.

$\begin{array}{rcl} |A - λ I| & = & 0 \\ |\begin{matrix} 0 - λ & 0 & 1 \\ 0 & 1 - λ & 0 \\ 1 & 0 & 0 - λ \end{matrix}| & = & 0 \\ |\begin{matrix} - λ & 0 & 1 \\ 0 & 1 - λ & 0 \\ 1 & 0 & - λ \end{matrix}| & = & 0 \\ - (1 - λ) [- λ (- λ) - 1] & = & 0 \\ - (1 - λ) [λ^{2} - 1] & = & 0 \\ - (1 - λ) [λ - 1] [λ + 1] & = & 0 \\ λ & = & - 1, 1, 1 \end{array}$

So, the eigenvalues are:

$λ = - 1, 1, 1$ .

Step 2

Now, find the eigenvectors.

$(A - λ I) X = O$

For, $λ = - 1$ ,

$\begin{array}{rcl} (A + I) X & = & O \\ [\begin{matrix} 1 & 0 & 1 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{matrix}] X & = & O \end{array}$

Apply,

$R_{3} \to R_{3} - R_{1} R_{2} \to R_{2} / 2$

$\begin{array}{rcl} [\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] & = & O \end{array}$

So, the equations are:

$\begin{array}{rcl} x_{2} & = & 0 \\ x_{1} + x_{3} & = & 0 \end{array}$

Suppose, $x_{3} = 1$ .

So,

$\begin{array}{rcl} x_{1} + 1 & = & 0 \\ x_{1} & = & - 1 \end{array}$

So, the eigenvector is:

$= [\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}]$ .

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