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Determene flo eigeuvedt or r of the matrix A. -4 A=

Determene flo eigeuvedt or r of the matrix A. -4 A=

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Delermine flo eraeuvect or of the matrix A= -7 -3
Transcribed Image Text:Delermine flo eraeuvect or of the matrix A= -7 -3
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A=\begin{pmatrix}0&-4&-7\\ 1&5&7\\ -4&-4&-3\end{pmatrix}

(A-xI) \:\:is,

\begin{pmatrix}0&-4&-7\\ 1&5&7\\ -4&-4&-3\end{pmatrix}-x\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{pmatrix}

= \begin{pmatrix}-x&-4&-7\\ 1&5-x&7\\ -4&-4&-3-x\end{pmatrix}

R_2\:\leftarrow \:R_2+\frac{1}{x}\cdot \:R_1

R_3\:\leftarrow \:R_3-\frac{4}{x}\cdot \:R_1

=\begin{pmatrix}-x&-4&-7\\ 0&\frac{5x-x^2-4}{x}&\frac{7x-7}{x}\\ 0&\frac{-4x+16}{x}&\frac{-3x-x^2+28}{x}\end{pmatrix}

R_2\:\leftrightarrow \:R_3

=\begin{pmatrix}-x&-4&-7\\ 0&\frac{-4x+16}{x}&\frac{-3x-x^2+28}{x}\\ 0&\frac{5x-x^2-4}{x}&\frac{7x-7}{x}\end{pmatrix}

R_3\:\leftarrow \:R_3-\frac{x-1}{4}\cdot \:R_2

=\begin{pmatrix}-x&-4&-7\\ 0&\frac{-4x+16}{x}&\frac{-3x-x^2+28}{x}\\ 0&0&\frac{\left(x-1\right)\left(x+3\right)}{4}\end{pmatrix}

now matrix is in triangular form

hence determinant is equal to the multiplication of diagonal entry

\left(-x\right)\left(\frac{-4x+16}{x}\right)\left(\frac{\left(x-1\right)\left(x+3\right)}{4}\right)=0

\left(-4x+16\right)\left(\frac{\left(x-1\right)\left(x+3\right)}{4}\right)=0

\left(-4\left(x-4\right)\right)\left(\frac{\left(x-1\right)\left(x+3\right)}{4}\right)=0

\left(x-4\right)\left(x-1\right)\left(x+3\right)=0

x=1,\:x=-3,\:x=4.................eigenvalues

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