Let P, Q be square matrices of order n and let 11,1, 11,2, . . ., u," be columns vectors in R\". Which of the following statements are always true? (There may have multiple answers.) C] If P, Q are orthogonal, then PQ is also orthogonal. C] If P, Q are orthogonal, then P+ Q is also orthogonal. C] If {11.1, 11.2, . . . , u" } i5 orthonormal, then (11,1 11.2 -- - u\") is an orthogonal matrix. C] If {11.1, uz, ..., un } is orthogonal, then (11.1 11,2 an) is an orthogonal matrix. Which of the following are eigenvectors of the matrix below? 2 0 0 1 1 *1 1 1 1 (There may have multiple answers.) Let A be a square matrix which has an eigenvector u associated with an eigenvalue 1. Which one of the following statements is false? O u is an eigenvector of A associated with 12. O If c is a nonzero real number, then cu is an eigenvector of A associated with cl. O u is an eigenvector of A + A2 associated with 1 + 12. O If A is invertible, then ) * 0 and u is an eigenvector of A associated withWhich one of the following square matrices is not diagonalizable? O 2 0 H O O O 2 O O O O 2 H O O 0 O 2 1 1 0 O O 1 O 0 O 0 0 O 0 O O O1 2 1 Let A beasquare matrix of order 3 and P: 1 0 1 1 1 1 1 0 0 Suppose P'IAP = 0 1 U 0 0 1 Which one of the following statements is false? 0 (1, 1, 1) is an eigenvector of A associated with the eigenvalue 1. Q dim(E,1) : 1 where E11 is the eigenspace of A associated with the eigenvalue 1. O dim(E1) : 1 where E1 is the eigenspace of A associated with the eigenvalue 1. O The characteristic equation of A is A3 + A2 7 A 7 1. 0 (1,1, 1) is an eigenvector of A associated with the eigenvalue 1. Let A, B be diagonalizable matrices of the same order. Which of the following statements are always true? (There may have multiple answers.) O If there exists an invertible matrix P which diagonalizes both A and B, then A - B is diagonalizable. O AB is diagonalizable. O If A is invertible, then A- is diagonalizable. O A + B is diagonalizable