Q33E A. Find the characteristic polyn... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 7: Q33E (page 337)

A. Find the characteristic polynomial of the matrix
$A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$
B. Can you find a $3 脳 3$ matrix whose characteristic polynomial is $- 位^{2} + 17 位^{2} - 5 位 + 蟿蟿$

Short Answer

Expert verified

Thus for the given matrix $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$ , the characteristic polynomial is $- 位^{3} + c 位^{2} - b 位 + a$ , and

the matrix for the given polynomial $- 位^{3} + 17 位^{2} - 5 位 + 蟺$ is $M = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 蟺 & - 5 & 17 \end{matrix}]$

Step by step solution

Finding the characteristic Polynomial for the given matrix.

Consider an matrix A and a scalar $位$ . Then $位$ is the eigenvalue of A if $\det (A - 位濒) = 0$ . This is called Characteristic equation of matrix A.

For the given matrix $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$ , the characteristic polynomial is

$\det (A - 位濒) = |\begin{matrix} - 位 & 1 & 0 \\ 0 & - 位 & 1 \\ a & b & c - 位 \end{matrix}| = 位^{2} (c - 位) + 产位 + a = - 位^{3} + {肠位}^{2} + 产位 + a$ ,

Finding the Matrix for the given polynomial.

In the part (B) , we need to find the $3 脳 3$ matrix whose characteristic polynomial is $- 位^{2} + 17 位^{2} - 5 位 + 蟺$ .

By comparing this with the part , we can get . So the matrix will be

$M = |\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 蟺 & - 5 & 17 \end{matrix}|$

Thus for the given matrix $A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$ , the characteristic polynomial is and the matrix for the given polynomial $- 位^{3} + 17 位^{2} - 5 位 + 蟺$ is $M = |\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 蟺 & - 5 & 17 \end{matrix}|$

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91影视

A. Find the characteristic polynomial of the matrix
$A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$
B. Can you find a $3 脳 3$ matrix whose characteristic polynomial is $- 位^{2} + 17 位^{2} - 5 位 + 蟿蟿$

Short Answer

Step by step solution

Finding the characteristic Polynomial for the given matrix.

Finding the Matrix for the given polynomial.

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91影视

A. Find the characteristic polynomial of the matrixA=[010001abc]B. Can you find a 3脳3matrix whose characteristic polynomial is-位2+17位2-5位+蟿蟿

Short Answer

Step by step solution

Finding the characteristic Polynomial for the given matrix.

Finding the Matrix for the given polynomial.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Calculus

Applied Mathematics

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.

A. Find the characteristic polynomial of the matrix
$A = [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ a & b & c \end{matrix}]$
B. Can you find a $3 脳 3$ matrix whose characteristic polynomial is $- 位^{2} + 17 位^{2} - 5 位 + 蟿蟿$