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TRUE/FALSE QUESTIONS FOR MIDTERM 2

MATH 54 TRUE/FALSE QUESTIONS FOR MIDTERM 2 SOLUTIONSPEYAM RYAN TABRIZIAN1. (a)TRUEIfAis diagonalizable, thenA3is diagonalizable.(A=PDP 1, soA3=PD3P= P D P 1, where P=Pand D=D3, which is diagonal)(b)TRUEIfAis a3 3matrix with3(linearly independent)eigenvectors, thenAis diagonalizable(This is one of the facts we talked about in lecture, the point isthat to figure out ifAis diagonalizable, look at the eigenvec-tors)(c)TRUEIfAis a3 3matrix with eigenvalues = 1,2,3,thenAis invertible(No eigenvalue which is0, so by the IMT,Ais invertible)(d)TRUEIfAis a3 3matrix with eigenvalues = 1,2,3,thenAis (always)

MATH 54 TRUE/FALSE QUESTIONS FOR MIDTERM 2 SOLUTIONS 3 Which gives y= 0, so y= 0, hence: Nul(I A) = ˆ x 0 ˙ = Span ˆ 1 0 ˙ Since there is only one (linearly independent) eigenvector, Ais not diagonalizable! 3. 1. (30 points, 5 pts each) Label the following statements as T or F. Make sure to JUSTIFYYOURANSWERS!!! You may use any

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