You are given the matrix A = 1 -2 -2 -1 -1 1 a. Find an expression for the determinant of the matrix A-l where A is a scalar and I is the 3x3 Identity Matrix. Your expression must contain at least one common factor. Do not expand into a cubic expression. b. Hence compute |A|. c. State the Characteristic Equation for matrix A. d. Hence identify the eigenvalues of matrix A. e. Find the eigenvector corresponding to any one of the eigenvalues.

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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You are given the matrix A = 1
-2
-2
6.
-1
-1
a. Find an expression for the determinant of the matrix A -Al where A is a scalar and I is the 3x3
Identity Matrix. Your expression must contain at least one common factor. Do not expand into
a cubic expression.
b. Hence compute |A|.
c. State the Characteristic Equation for matrix A.
d. Hence identify the eigenvalues of matrix A.
e. Find the eigenvector corresponding to any one of the eigenvalues.
f. Carefully describe the approach which you will take to diagonalize the matrix A.
g. Compute A? i.e. the square of matrix A.
h. Hence create the quadratic form G(x) = x' Bx where B = A? and x = [x1, x2, x3].
1. State the eigenvalues of the matrix A?.
J. Hence classify the quadratic form G(x) in terms of its definiteness. Justify your answer.
k. Carefully describe the approach which you will take to convert the quadratic form G(x)
1-1 D D90%
Transcribed Image Text:You are given the matrix A = 1 -2 -2 6. -1 -1 a. Find an expression for the determinant of the matrix A -Al where A is a scalar and I is the 3x3 Identity Matrix. Your expression must contain at least one common factor. Do not expand into a cubic expression. b. Hence compute |A|. c. State the Characteristic Equation for matrix A. d. Hence identify the eigenvalues of matrix A. e. Find the eigenvector corresponding to any one of the eigenvalues. f. Carefully describe the approach which you will take to diagonalize the matrix A. g. Compute A? i.e. the square of matrix A. h. Hence create the quadratic form G(x) = x' Bx where B = A? and x = [x1, x2, x3]. 1. State the eigenvalues of the matrix A?. J. Hence classify the quadratic form G(x) in terms of its definiteness. Justify your answer. k. Carefully describe the approach which you will take to convert the quadratic form G(x) 1-1 D D90%
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