Science:Math Exam Resources/Courses/MATH221/December 2007/Question 07 (c)
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Question 07 (c) 

Consider the matrix . For each eigenvalue, find the dimension of the corresponding eigenspace. 
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Hint 

Science:Math Exam Resources/Courses/MATH221/December 2007/Question 07 (c)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From part (b), we know that the eigenvalues of A are 1 and 2. We also saw in part (b) that the eigenvalue 2 is a repeated root of the characteristic polynomial for A, while the eigenvalue 1 is not. By definition, the dimension of each respective eigenspace is at least 1. As well, recall that the dimension of each eigenspace is at most the multiplicity of the eigenvalue as a root of the characteristic polynomial.
To calculate the dimension of the eigenspace corresponding to the eigenvalue 2, we want to determine , that is, the dimension of the kernel of A  2I. We have
All three rows of A  2I are multiples of each other, so . Therefore, by the ranknullity theorem,
In conclusion,
