Suppose that A is a 7 x 7 matrix whose only distinct eigenvalues are 1, 2, and 3. Let cA be the characteristic polynomial of A. The degree of CA(x) is The fewest basic 1-eigenvectors that A can have is and the most basic 1-eigenvectors that A can have is Now, suppose in addition that A is diagonalizable and that A has three basic 1-eigenvectors. Under these assumptions, fill in the blanks in the exponents of CA(x) below, and determine the five ranks listed below. 1)0(x CA(x) = (x-1) (x - 2)²(x-3) rank(I7- A) = rank (217 - A) = rank (317- A) = | rank (417 - A) = rank(A) =

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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Suppose that A is a 7 x 7 matrix whose only distinct eigenvalues are 1, 2, and 3. Let cà be the characteristic polynomial of A.
The degree of CÂ(x) is
The fewest basic 1-eigenvectors that A can have is
and the most basic 1-eigenvectors that A can have is
Now, suppose in addition that A is diagonalizable and that A has three basic 1-eigenvectors. Under these assumptions, fill in the blanks in the exponents
of CA (x) below, and determine the five ranks listed below.
€₁(x) = (x − 1)(x − 2)²(x − 3)
rank(I7 - A) =
rank(217 — A) =
rank(317 — A) =
=
rank (417 - A)
=
rank(A) =
Transcribed Image Text:Suppose that A is a 7 x 7 matrix whose only distinct eigenvalues are 1, 2, and 3. Let cà be the characteristic polynomial of A. The degree of CÂ(x) is The fewest basic 1-eigenvectors that A can have is and the most basic 1-eigenvectors that A can have is Now, suppose in addition that A is diagonalizable and that A has three basic 1-eigenvectors. Under these assumptions, fill in the blanks in the exponents of CA (x) below, and determine the five ranks listed below. €₁(x) = (x − 1)(x − 2)²(x − 3) rank(I7 - A) = rank(217 — A) = rank(317 — A) = = rank (417 - A) = rank(A) =
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