The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows.A²6λ + 11 = 01 -3A =and by the theorem you have2 5A² - 6A + 111₂ = 0Demonstrate the Cayley-Hamilton Theorem for the matrix A given below.0 2 -1A =-1 4 10 0 -1STEP 1: Find and expand the characteristic equation.STEP 2: Compute the required powers of A.A²A³ =100100LOODC STEP 3: Write a matrix version of the characteristic equation by replacing with A. (Use I for the 3x3 identity matrix.)STEP 4: Substitute the powers of A into the matrix equation from step 3, and simplify. Is the matrix equation true?YesNo

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Answered: The Cayley-Hamilton Theorem states that…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CR: Review Exercises

Problem 64CR: a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving...

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a Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.
Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].
A square matrix is called upper triangular if all of the entries below the main diagonal are zero. Thus, the form of an upper triangular matrix is where the entries marked * are arbitrary. A more formal definition of such a matrix . 29. Prove that the product of two upper triangular matrices is upper triangular.
Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.
35. A permutation matrix is a matrix that can be obtained from an identity matrix by interchanging the rows one or more times (that is, by permuting the rows). For the permutation matrices are and the five matrices. (Sec. , Sec. , Sec. ) Given that is a group of order with respect to matrix multiplication, write out a multiplication table for . Sec. 22. Find the center for each of the following groups . c. in Exercise 35 of section 3.1. 32. Find the centralizer for each element in each of the following groups. c. in Exercise 35 of section 3.1 Sec. 5. The elements of the multiplicative group of permutation matrices are given in Exercise of section. Find the order of each element of the group. Sec. 6. Let be the group of permutations matrices as given in Exercise of Section .
Let A,D, and P be nn matrices satisfying AP=PD. Assume that P is nonsingular and solve this for A. Must it be true that A=D?
12. Positive integral powers of a square matrix are defined by and for every positive integer. Evaluate and and compare the results for and.

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