From the 8 choices, select all that applySuppose A is an n x n matrix.Which of the following statements must be true?A has at least one eigenvector.If A has an eigenvector, then A has infinitely many eigenvectors.If A has n distinct eigenvalues, then A is diagonalizable.If A is diagonalizable, then A has n distinct eigenvalues.If A is diagonalizable, then A223 is diagonalizable.If the system Ax = ₁, has infinitely many solutions, then 0 is an eigenvalue of A. (Recall that e₁ is the vector with a 1 in its first coordinate and Os in all theothers).If A has exactly two basic 7-eigenvectors, then rank(7In − A) = 2.If n is odd, then A has at least one (real) eigenvalue.

Given: Suppose A is an n x n matrix.

Answered: he 8 choices, select all that apply…

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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? ? ? Are the following statements true or false for a square matrix A? 1. The sum of two eigenvectors of a matrix is always an eigenvector. 2. If an n x n matrix A is diagonalizable, then A has n distinct eigenvalues. 3. If A is invertible, then A is diagonalizable.
Find an invertible matrix P and a matrix C of the form O B. A. The matrices P and C are 0. (Use a comma to separate answers as needed.) There is no matrix C of the form eigenvalues of A are 3-i and 3 + i. The corresponding eigenvectors are v₁ = respectively. a a b b - b a - b a such that A= 1 -5 1 ਜ਼ਿੰ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 5 - 2-i 1 has the form A = PCP¯1. The and v₂ = and no invertible matrix P such that A = PCP¯ 1. - 2+i 1
Select T if the statement is true (in all cases), or select F if false (for at least one example).
Which of the following is true? A Qd AT have different eigen values. Q If astA = detB, then A and B are similar. An eigenspace of A is a null space of a certain matrix. All the eigenvectors corresponding to i is called eigenspace of A corresponding to 2. أخل اختياري
Let A be an n xn real matrix with det( A) + 0. Which of the following statements is false? Select one: O a. All eigenvalues of A are nonzero. O b. The inverse matrix A-1 exists. O c. The rank of A is n. O d. There exist infinitely many solutions to the system Ax = b, for any vector b € R". %3D
The two that are checked are true for sure, but I can't find where the textbook mentions anything about the others. Thank you.
i. Show that the matrix is not diagonalizable. ii. Find the eigenvalues of the matrix and determine whether there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable. -3 -2 3 3 4 -9 1. 2 -5)
b. Which of the following statements are true? Select all that apply. A. The matrices A and AT have the same eigenvalues, counting multiplicities. B. The sum of two eigenvectors of a matrix A is also an eigenvector of A. | C. Each eigenvalue of A is also an eigenvalue of A?. D. Each eigenvector of A is also an eigenvector of A?. E. Each eigenvector of an invertible matrix A is also an eigenvector of A1.
Prove that the symmetric matrix is diagonalizable. (Assume that a is real.) 00a --[6] A = 0 a 0 a 00 Find the eigenvalues of A. (Enter your answers as a comma-separated list. Do not list the same eigenvalue multiple times.) λ = Find an invertible matrix P such that P-¹AP is diagonal. P = Which of the following statements is true? (Select all that apply.) O A is diagonalizable because it has 3 distinct eigenvalues. OA is diagonalizable because it has 3 linearly independent eigenvectors. O A is diagonalizable because it is a symmetric matrix. A is diagonalizable because it has a nonzero determinant. A is diagonalizable because it is an anti-diagonal matrix. A is diagonalizable because it is a square matrix. OA is diagonalizable because it has a determinant of 0.
Let A be an invertible 3x3 matrix. Mark all that are necessarily true about this matrix: The dimension of the vector space spanned by the columns of A is at most 2. The columns of A are linearly independent. O A is diagonalizable O O is not an eigenvalue of A.
Let A be a symmetric and positive definite matrix. Select all true statements I. All eigenvalues of A are positive. II. A can be reduce to upper triangular matrix form using only row operations type II and the pivot elements will all be positive. II. We cannot find the sign of det(A). A. I, II B. I, II O C.I, II, II D. II, II
I. Answer "True" or "False" for each statement. If the statement is "true," provide justification (proof) your answer. If the statement is "false," provide a counter example that would be true. for i. If i is in both the nullspace and the column space of an n x n matrix A, then i = ỡ. ii. If A is a square matrix, NS(A) is the same as the eigenspace Eo · ii. If A is a square matrix, every vector is an eigenvector in the eigenspace E·

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