Suppose A is a 5 by 5 matrix and the reduced echelon form of A is U.Decide whether each of the following statements is true.Statement 1:A and U must have the same rank.Statement 2:If v is an eigenvector of A with eigenvalue 0 then v must also be an eigenvector of U with eigenvalue 0.Select one alternative© Statement 1 is false and statement 2 is trueStatement 1 is false and statement 2 is falseO Statement 1 is true and statement 2 is trueStatement 1 is true and statement 2 is falseO

Answered: Image /qna-images/answer/1489e89c-b17f-49f5-8619-410f6d3b84e7.jpg

Answered: Suppose A is a 5 by 5 matrix and the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that A is square matrix, prove that: • the sum of its eigenvalues is equal to the trace of A; • the product of its eigenvalues is equal to the determinant of A.
If matrix A is square of size n x n with a rank equal to 2 and n > 2. The rank of a matrix is equal to the number of non-zero eigenvalues. Can you write A as the product of two matrices B E Rnxq and C E RPX" where we require that p < n and q < n? You may assume that A is diagonalizable. Show how to construct a matrix B and C such that A = BC.
Let A be an 3 by 3 matrix. Select all true statements below. A. If A is diagonalizable, then A has 3 distinct real eigenvalues. B. If A has 3 linearly independent eigenvectors, then A is diagonalizable. C. The matrix A may or may not be diagonalizable. D. The matrix A is certainly diagonalizable. E. If A has 3 distinct real eigenvalues, then A is diagonalizable. OF. If A is diagonalizable, then A has 3 linearly independent eigenvectors. G. None of the above
Suppose a matrix A has eigenvalues -1,-2 and 3. Then the eigenvalues of A3+/ are: 9,2 and 28. O -9,-2 and 26. 0,-7 and 28. None of these. o 1,3 and 16.
Suppose that a 5 x 5 matrix over C has no eigenvectors. What are the possible structures for its Jordan form up to ordering of blocks? Select one: O 51-blocks or 3 1-blocks and a 2-block or 2 1-blocks and a 3-block or a 1-block and a 4-block or just 1 5-block O There is not enough information to determine this as it depends on the entries of the matrix O None of the others apply O No Jordan blocks because there are no eigenvectors and one needs at least one eigenvalue to construct a Jordan block either a 5-block or one 2-block and one 3-block
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·
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
If B and C are positive matrices, show that A has a positive real eigenvalue r with the property that |λ| < r for any eigenvalue λ ≠= r. Show also that the multiplicity of r is at most 2 and that r has a nonnegative eigenvector.

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