Transcription of TRUE/FALSE QUESTIONS FOR MIDTERM 2
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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) diagonalizable(this is the useful test we ve been talking about in lecture,Aisdiagonalizable since it has3distinct eigenvalues)(e)FALSEIfAis a3 3matrix with eigenvalues = 1,2,2,thenAis (always) not diagonalizable(TakeA= 1 0 00 2 00 0 2 , it is diagonal, hence diagonalizable)Date: Monday, April 13th, RYAN TABRIZIAN(f)FALSEIf xis the orthogonal projection ofxonW, then xisorthogonal tox.
MATH 54 TRUE/FALSE QUESTIONS FOR MIDTERM 2 SOLUTIONS 5 b( 1 2) = 0 But 1 6= 2, so 1 2 6= 0 , hence we get b= 0. But going back to the first equation, we get: av 1 = 0 So a= 0. Hence a= b= 0, and we’re done! (d) If a matrix Ahas orthogonal columns, then it is an orthogonal matrix. FALSE Remember that an orthogonal matrix has to have ...
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