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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A is an n x n matrix. Check the true statements below. A. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy. B. A matrix A is not invertible if and only if 0 is an eigenvalue of A. C. A number is an eigenvalue of A if and only if the equation (A - cI)z = 0 has a nontrivial solution D. To find the eigenvalues of A, reduce A to echelon form. E. If Az = Az for some vector , then A is an eigenvalue of A.
Select T if the statement is true (in all cases), or select F if false (for at least one example).
Let A be a symmetric matrix with eigenvalues 0 and 1 and corresponding eigenvectors (a, b) and (c, d). a c If the eigenvector matrix is S= then the eigenvectors of B = S-1 A S are a. cannot be determined O b. (3, 3), (-3, -3) O. (2, 2), (-2, -2) O d. (1, 0), (0, 1) O e. (2, ), (-2, -i)
Find orthogonal matrices that diagonalized the matrix below. Show all work computing eigenvalues and eigenvectors. Write the factorization of the positive definite matrix below. 2 9
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.
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.
Which of the following is true? Select all possible answers. If we square an arbitrary matrix A, the eigenvalue of A2 can be computed from the eigenvalue of A The eigenvalues of an upper triangular matrix are its diagonal entries For any two matrices A and B, the eigenvalues of A and B are always the sum of the eigenvalues of A and B If U and V are two vector spaces, then the basis of their union is the union of the basis sets of U and V
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 ? be a 3×3 matrix and let ?⃗ 1, ?⃗ 2 be eigenvectors for ?. Further, assume ?⃗ 1 and ?⃗ 2 are linearly independent.
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

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