Truefalse - I . (12) If A and B have the same eigenvalues...

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MATH 343:001 (11814) Applied Linear Algebra Fall 2009 True/False questions for class discussion (1) If A and B are square matrices with the same characteristic polynomial then they are similar. (2) If A is invertible and symmetric then A 2 is positive definite. (3) The matrices 2 0 0 - 3 and 2 5 0 - 3 are similar. (4) If Q is unitary then its eigenvalues are real. (5) If A satisfies A 2 = A then the only eigenvalues of A are 0 and 1. (6) If A is symmetric it eigenvalues are real. (7) If A is skew-symmetric its eigenvalues are real. (8) The eigenvalues of AB are the eigenvalues of A multiplied by the eigenvalues of B . (9) If A and B commute, A is nonsingular and λ is an eigenvalue of A then it is also an eigenvalue of B . (10) If A is positive definite then - A is positive definite. (11) The only positive definite projection matrix is
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Unformatted text preview: I . (12) If A and B have the same eigenvalues and eigenvectors then A = B . (13) If A is similar to B and B is similar to C then A is similar to C . (14) If A is an m n matrix, then A and A T have the same nullity. (15) If A and B are n n matrices then det( AB ) = det( BA ). (16) If A is Hermitian, then A + iI is invertible. (17) If U is unitary, then U + 2 I is invertible. (18) An invertible matrix cant be similar to a singular matrix. (19) If A and B are square matrices then trace( AB ) = trace( BA ). (20) If A is similar to B then e A is similar to e B . Gerardo Laerriere, November 30, 2009...
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