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In this lesson, we will learn how to find the eigenvalues and eigenvectors of a matrix and how to determine whether a matrix is defective or not.

Q1:

For the matrix find the eigenvalues and eigenvectors and determine whether it is defective.

Q2:

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

Q3:

Q4:

Q5:

Q6:

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

Q7:

Q8:

Q9:

Q10:

Q11:

Q12:

Q13:

Q14:

Find the eigenvectors of the matrix

Q15:

Let 𝑇 be the linear transformation that rotates all vectors in ℝ 2 counterclockwise through an angle of 𝜋 2 . Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

Q16:

Find the eigenvalues and eigenvectors of the matrix where and are real numbers.

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