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Posted: August 9th, 2023
The characteristic polynomial of A = 7 0 -105 2 -105 0 -8is -(t+3)(t-2)^2I got that the eigenvalues are t = -3 multiplicity 1, and t = 2 multiplicity 2.For each eigenvalue find a basis for the corresponding eigenspace.Then write the matrices P and D for a diagonalization of A.I think I know how to do this, but evidently I don’t :P.In the case of t = -3, I added 3 to the values along the diagonal, then got to rref and found that the first to columns were pivot columns, telling me that the first two columns of the original matrix formed the eigenspace.For t = 2, I subtracted 2 from the values along the diagonal, got to rref and found that there as only one pivot column, telling me that the first column of the original matrix forms the eigenspace.That got marked wrong, so I guess that’s wrong… but the professor didn’t feel like explaining why.Why?Also, for the matrices P and D for a diagonalization of A, I gotD:-3 0 00 2 00 0 2And for P:0 7 72 5 50 5 5
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