This document contains 10 true/false statements about properties of matrices and eigenvalues. It states that (i) doubling the matrix doubles the eigenvalues, (ii) eigenvectors of A and A^2/3 correspond to the same eigenvalue, (iii) the rank is less than or equal to n if A is an eigenvalue, (iv) there can be a matrix with 0 as an eigenvalue and full column rank, (v) linearly independent columns means 0 cannot be an eigenvalue, (vi) eigenvectors of a symmetric matrix for distinct eigenvalues are orthogonal, (vii) a diagonalizable matrix has a unique transformation matrix to diagonal form, (viii) the property of being diagonalizable is preserved under taking powers, and (x)

1. 10. [10 points] True or False. For all subquestions below, assume that A is an n x n matrix. (i) T
F: a is an eigenvalue of A if and only if 2 is an eigenvalue of 2A (ii) T F: Suppose A and A2 1/3
are two distinct eigenvectors of A. Then x is an eigenvector corresponding to the eigenvalue A if
and only if 3x is an eigenvector corresponding to the eigenvalue A2. (iii) T F: I f A is an n x n
matrix and A is one of its eigenvalues, then rank(AIn A) n (iv) T F: There is an n x n matrix A
such that A 0 is an eigenvalue of A and Col (A) Rn (v) T F: If the column vectors of a square
matrix A are linearly independent, then 0 is not an eigenvalue of A (vi) T F: Two eigenvectors of
a symmetric matrix A corresponding to two distinct eigenvalues are orthogonal to each other
(vii) T F: If a square matrix A is diagonalizable, then there is a unique matrix P such that P-IAP
is diagonal. (viii) T F: I f a square matrix A is diagonalizable, then so is A (ix) T F: I A is an
eigenvalue of a square matrix A, then Ak must be an eigenvalue of A for any f positive integer k.
(x) T F: T column vectors u and v in C" are complex orthogonal if and only if u T. -0 wo
Solution
1) TRUE
2)TRUE
3)FALSE
4)TRUE
5) FALSE
6) TRUE
7)TRUE
8)TRUE
9)FALSE
10) TRUE