Science:Math Exam Resources/Courses/MATH152/April 2012/Question 04 (c)
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Question 04 (c)
Consider the matrix
Use the information in items (a) and (b) to write A as , where M is an invertible matrix and D is a diagonal matrix.
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The columns of M correspond to the eigenvectors and the values of D are the eigenvalues in order of the eigenvectors in the matrix M.
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The matrix M is made up of the eigenvectors of A and so
The matrix D is a diagonal matrix of the eigenvalues listed in the order that we list the eigenvectors in M. Therefore,
We can also compute by recalling that for a 2x2 matrix Q such that
Since det(M)=2, we have that
Multiplying everything together we get
as we were expecting.