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# Calc III for Comput Sci MATH 2605

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This 0 page Class Notes was uploaded by Chelsea Nolan MD on Monday November 2, 2015. The Class Notes belongs to MATH 2605 at Georgia Institute of Technology - Main Campus taught by John Elton in Fall. Since its upload, it has received 19 views. For similar materials see /class/233951/math-2605-georgia-institute-of-technology-main-campus in Mathematics (M) at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Here are some problems on eigenvalsVects and determinants taken from old nals or exams or practice material There may be another few added later 9 4 0 lLetA 6 l 0 6 4 3 a Find the eigenvalues of A be smart and expand the determinant by the last column For your information the eigenvalues for this matrix are integers and one of them is 3 Please do the algebra carefully b Find a basis for ker A 7 31 What is the dimension of this kernel you can do paIt b even if you did not do paIt a This kernel is called the eigenspace associated with the eigenvalue 3 and all nonzero elements in it are eigenvalues The dimension is called the geometric multiplicity of the eigenvalue 1 3 2 Let A I Find the eigenvalues of A Find corresponding eigenvectors of A Find a matrix V and a diagonal matrix D such that V39lAV D you are not required to multiply it out unless you just want to as a check Find V391 Find a formula for A1 that is just 3 matrices multiplied together you do not have to actually multiply them 2 1 1 3 Let A l 3 0 Given that 7 l is an eigenvalue of A you do not have to find any of the 1 1 2 other eigenvalues of A nd an eigenvector associated with this eigenvalue 1 1 2 4 Let A 2 I Find the eigenvalues of A they are integers Find the corresponding eigenvectors of A Find a matrix V and a diagonal matrix D such that V39lAV D you are not required to multiply it out unless you just want to as a check Express this as A VDV39l Use this to nd a formula for An written as the product of 3 matrices Note that the eigenvectors are 01thogonal for this problem that happens whenever the matrix A is symmetric but not in general So if you normalize the eigenvectors your V will satisfy VtV I so then Vt V39l so we don t have to do anything to compute the inverse matrix That is useful for large matrices but it hardly matters for a 2x2 matrix 5 Find the volume ofthe parallelepiped with edges 13 1 0 2 l 1 l 2

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