True or False? If A is a 3x3 matrix with characteristics
polynomial p(lamda) = (lamda)^3-(lamda)^2+2(lamda), then A is
invertible.
True or False? If A is a 3x3 matrix with characteristics polynomial p(lamda) = (lamda)^3-(lamda)^2+2(lamda), then...
True or false. Please justify
why true or why false also
(I) A square matrix with the characteristic polynomial 14 – 413 +212 – +3 is invertible. [ 23] (II) Matrix in Z5 has two distinct eigenvalues. 1 4 (III) Similar matrices have the same eigenspaces for the corresponding eigenvalues. (IV) There exists a matrix A with eigenvalue 5 whose algebraic multiplicity is 2 and geo- metric multiplicity is 3. (V) Two diagonal matrices D1 and D2 are similar if...
Problem #30: [2 marks] Suppose that a matrix A has characteristic polynomial p() = 1 - 31' + 814 - 23. Consider the following statements. (i) i = 2 is an eigenvalue of A. (ii) A is a 4 x 4 matrix. (iii) That same p() is also the characteristic polynomial of A! Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True...
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
If W = Col A where A is a 3x3 matrix having rank = 2 , it is possible that W! is a plane in R3 containing the origin. True False If W = span{x1, #2, 3} and {x1, #2, #3} is an independent set, and if {ū1, 72, 73 } is an orthogonal set in W then 9 {ū1, U2, U3 } is an orthogonal basis for W True False
Part A. (True/False Questions) (15 pts). Decide if the given statement is true or false. (Justify briefly your answer) 1. The eigenvalues of the matrix A = -5 6 are: 5 and -4. O True False 2. Let A= 2 -4 be a square matrix. The vector v= [ is an eigenvector of the matrix A. 2 True False 3. If I = -4 is an eigenvalue of a 5 x 5 matrix A, then Av = -4v for any...
True or False? Justify your answer. Answers without correct
justification will receive no credit.
(I) A square matrix with the characteristic polynomial λ 4 −4λ 3
+ 2λ 2 −λ+ 3 is invertible.
(II) Matrix 2 3 14 in Z5 has two distinct eigenvalues.
Indicate whether the statement is true or false: If a matrix is invertible and diagonalizable, then its inverse is diagonalizable O O True False
Exercise 1 Let the matrix 15 00 A-010 20 -3 a) Find an invertible matrix P such that P-TAP is diagonal. b) Find the minimal polynomial. c) Find 410 (Note that 30 = 59049).
True or False: If A is an matrix that is both diagonalizable and invertible, then so is A-1. If true, briefly explain why; if false give a counterexample. Hint: consider taking the inverse of both sides of the equation A = PDP-1
(b) In each case below, state whether the statement is true or false. Justify your answer in each case. (i) A+B is an invertible 2×2 matrix for all invertible 2×2 matrices A, B. [4 marks] (ii) If A is an n×n invertible matrix and AB is an n×n invertible matrix, then B is an n × n invertible matrix, for all natural numbers n. [4 marks] (iii) det(A) = 1 for all invertible matrices A that satisfy A = A2....