Let A be a square matrix. Which of the following statements are always true? i) if A is an eigenvalue for A, then -A is an eigenvalue for -A ii) if V is an eigenvector for A, then V is also an eigenvector for 2A iii) The eigenspace for the matrix A for the eigenvalue A has dimension equal to the multiplicity of A i) only ii) and iii) only i) and ii) only i) and iii) only 1), ii) and ili)

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

Consider the following matrix: A = ..comm 3 0 0 00 03 0 -5 0-12 -3 12 000-2 a) Find the distinct eigenvalues of A, their multiplicities, and the corresponding number of basic eigenvectors. Number of Distinct Eigenvalues: 1 Eigenvalue: 0 has multiplicity 1 and corresponding number of basic eigenvectors 1 b) Determine whether the matrix A is diagonalizable. Conclusion: A is diagonalizable Question 13 A is diagonalizable A is not diagonalizable sk dom A in the diagonalizablo matrix boloy and P-1AP-D for

Write a formal proof for the given statements below. (b)  Let A be an n × n matrix. Suppose that vector v ∈ Rn is an eigenvector of A with the corresponding eigenvalue λ ∈ R. Prove that vector v is an eigenvector of A3 with the corresponding eigenvalue λ3 .