MATH221 April 2009
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Question 12 (i)
There is no matrix A with eigenvectors , and with corresponding eigenvalues 1, -1, 4.
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The answer is false.
Let and denote the given vectors.
A matrix A has eigenvectors with corresponding respective eigenvalues 1, -1, 4 if and only if , where
Thus the matrix A exists if and only if P is invertible (that is, if and only if are linearly independent).
We can calculate . Therefore, P is invertible and the desired matrix A will be given by .
Remark: By computing the inverse (not necessary to answer the question), we can determine the explicit form of A as
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