Question

Q1. Let A = be a 2 x 2 matrix. 30 (a) Find the characteristic polynomial of the matrix A. (5 pts) (b) Find all eigenvalues and associated eigenvectors of the matrix A. (10 pts) (c) If is an eigenvalue of A, what do you think it would be the eigenvalue of the matrix 7A?(Justify your answer) (5 pts)

Answer 1

Please check out my solution.

Parts (a) and (b) are solved below.

SOLUTION Your input: find the eigenvalues and 4 eigenvectors of A = 30 -9 Start from forming a new matrix by subtracting from the diagonal entries of the given matrix: 4- -1 30 -1-9 Find the determinant of the obtained matrix: 4-1 -1 12 +51-6 = 30 -X-9

This is a characteristic polynomial. Solve the equation 12 + 51 – 6= 0. The roots are: 11 = 1 12 = -6 These are the eigenvalues. Next, find the eigenvectors. a. l=1 4-1 -1 30 -1-9 3 -1 = - 30 -10

Perform row operations to obtain the rref of the matrix: 3 30 -10 1 - 1 0 0. Now, solve the matrix equation - 1 JO-D LO If we take V2 = t, then t V2 = t. 3 V1 = Therefore, --[-U t t 1

b. = -6 4-2 -1 - -9 30 [10 1 [30 -3 Perform row operations to obtain the rref of the matrix: 1 10 -11 1 10 30 -3 0 0 Now, solve the matrix equation 1 01 10 0 0 02. 이 이 -

If we take V2 = t, then t V1 t. 10 = V2 = Therefore, t 10 10 V = t t 1 ANSWER COP Eigenvalue: 1, eigenvector: 1 Eigenvalue: –6, eigenvector: 10 1

Part (c) is solved below.

28 -7 A= 210 -63 Start from forming a new matrix by subtracting from the diagonal entries of the given matrix: 28 – 1 -7 210 -- 63 Find the determinant of the obtained matrix: 28 - -7 = 12 - 210 + 357 -1- 63 294

This is a characteristic polynomial. Solve the equation 12 + 352 – 294 = 0. The roots are: 21 = 7 } (eigenvalues) 12 = -42
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