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In this worksheet, we will practice finding the eigenvalues of a matrix and the corresponding eigenvectors of a matrix.

Q1:

Calculate the eigenvalues of π΄ = ο β 1 0 0 β 1 ο .

Q2:

Is it possible for a nonzero matrix to have 0 as its only eigenvalue?

Q3:

Can a real 3 Γ 3 matrix which has a nonreal eigenvalue be defective?

Q4:

Fill in the blanks.

Let π΄ be an π Γ π matrix. Then t r a c e ( π΄ ) equals and d e t ( π΄ ) equals .

Q5:

Suppose π β πΏ ( π , π ) is a linear operator. Then π β πΉ is an eigenvalue of π if and only if

Q6:

Suppose that π , π£ is an eigenpair of the invertible matrix π΄ . Then 1 π , π£ is an eigenpair of which of the following matrices?

Q7:

If π΄ is a 4 Γ 4 real matrix, which of the following is true of the matrix π΄ π΄ ο³ ?

Q8:

Find the eigenvalues of π΄ = ο 0 1 β 1 0 ο .

Q9:

Suppose 3 Γ 3 a matrix π΄ is row-equivalent to the identity matrix. Which of the following may be false?

Q10:

What is the characteristic polynomial of the matrix π΄ = ο 4 1 β 1 0 ο ?

Q11:

Consider a square matrix π΄ with eigenvalue π . Is it true that any vector in the nullspace of π΄ β π πΌ is an eigenvector of π΄ corresponding to π ?

Q12:

Suppose one eigenvalue of a real matrix π΄ is π = β 1 + 4 π ο§ and the corresponding eigenvector is π£ = ο 3 + 2 π β 1 ο . ο§ Which of the following must also be an eigenpair of π΄ ?

Q13:

Suppose that π£ is an eigenvector of a matrix π΄ corresponding to a nonzero eigenvalue π . Is π΄ π₯ = π£ always solvable?

Q14:

Suppose an π Γ π matrix π΄ is diagonalizable. Then, which of the following does π΄ always have?

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