Class Test M.Sc. Data Sci.

Calculate the eigenvalues of the matrix [[2, 1], [1, 2]].

• [1, 1]

• [4, 0]

• [3, 1]

• [2, 2]

Find the eigenvectors corresponding to the eigenvalue 3 for the matrix [[3, 0], [0, 2]].

• Eigenvectors corresponding to eigenvalue 3 are of the form k*[1, 0], where k is any non-zero scalar.

• Eigenvectors corresponding to eigenvalue 3 are of the form k*[0, 1], where k is any non-zero scalar.

• Eigenvectors corresponding to eigenvalue 3 are of the form k*[0, 0], where k is any non-zero scalar.

• Eigenvectors corresponding to eigenvalue 3 are of the form k*[1, 1], where k is any non-zero scalar.

Diagonalize the matrix [[4, 1], [2, 3]].

• The diagonalized form is D = [[7, 0], [0, 0]] with P = [[0, 1], [1, 0]].

• The diagonalized form is D = [[4, 0], [0, 3]] with P = [[1, 0], [0, 1]].

• The diagonalized form is D = [[5, 0], [0, 2]] with P = [[1, -1], [1, 2]].

• The diagonalized form is D = [[6, 0], [0, 1]] with P = [[1, 1], [1, 1]].

Apply the eigenvalue theorem to the matrix [[5, 0], [0, 5]]. What does it imply?

• The eigenvalue is 5 with multiplicity 1, indicating rotation.

• The eigenvalue is 10 with multiplicity 2, indicating reflection.

• The eigenvalue is 5 with multiplicity 2, indicating uniform scaling.

• The eigenvalue is 0 with multiplicity 2, indicating no scaling.

Solve the eigenvalues of a real symmetric matrix are

• purely real

• purely imaginary

• May be complex

• All of the abobe.

Calculate the eigenvalues of the matrix [[1, 2], [2, 1]].

• [1, 1]

• [0, 4]

• [3, -1]

• [2, 2]

Find the eigenvectors for the eigenvalue 1 of the matrix [[1, 2], [2, 1]].

• Eigenvectors are of the form [1, 1] for any non-zero scalar.

• Eigenvectors are of the form [t, t] for any t ≠ 0.

• Eigenvectors are of the form [t, 0] for any t ≠ 0.

• Eigenvectors are of the form [0, t] for any t ≠ 0.

Diagonalize the matrix [[0, 1], [-2, -3]].

• D = [[0, 1], [2, 3]]

• D = [[1, 0], [0, 2]]

• D = [[-1, 0], [0, -2]]

• D = [[-2, 0], [0, -1]]

Using the eigenvalue theorem, determine the stability of the system represented by the matrix [[0, 1], [-2, -3]].

• The system is stable.

• The system has complex eigenvalues.

• The system oscillates indefinitely.

• The system is unstable.

The is A is a singular 2x2 matrix having one of the eigen value is 2 then Trac(A) will be

• 2

• -2

• 0

• 4