Eigenvalues and eigenvectors; diagonalization of 2x2 practice questions.

Calculate the sum of the eigenvalues of the matrix M = [[6, 2], [1, 5]] (rows shown).

• 11
• 13
• 28
• 4

Answer: 11

The sum of the eigenvalues equals the trace: 6 + 5 = 11.

Determine the eigenvalues of the matrix [[2, 0], [0, 7]] (rows shown).

• 2 and 7
• 0 and 0
• 9 and 14
• −2 and −7

Answer: 2 and 7

For a diagonal (or triangular) matrix the eigenvalues are the entries on the main diagonal, so 2 and 7.

The characteristic equation of a 2×2 matrix M is λ² − 7λ + 12 = 0. State the eigenvalues of M.

• λ = 3 and λ = 4
• λ = −3 and λ = −4
• λ = 7 and λ = 12
• λ = 1 and λ = 12

Answer: λ = 3 and λ = 4

Factorising λ² − 7λ + 12 = (λ − 3)(λ − 4) = 0 gives λ = 3 and λ = 4.

Find an eigenvector corresponding to the eigenvalue λ = 5 of the matrix [[3, 2], [2, 3]] (rows shown).

• (1, 1)
• (1, −1)
• (2, 3)
• (5, 0)

Answer: (1, 1)

For λ = 5, (M − 5I)v = 0 gives [[−2, 2], [2, −2]]v = 0, so −2x + 2y = 0, i.e. y = x; the vector (1, 1) satisfies this.

A matrix M has eigenvalues λ₁ = 2 and λ₂ = 3. In the diagonalization M = PDP⁻¹, which of the following is a valid diagonal matrix D?

• [[2, 0], [0, 3]]
• [[2, 3], [0, 0]]
• [[5, 0], [0, 6]]
• [[3, 0], [2, 0]]

Answer: [[2, 0], [0, 3]]

D is the diagonal matrix whose diagonal entries are the eigenvalues, so [[2, 0], [0, 3]] (order matching the eigenvector columns of P).

Determine the value of the product of the eigenvalues of M = [[4, 3], [2, 1]] (rows shown).

• −2
• 2
• 5
• 10

Answer: −2

The product of the eigenvalues equals the determinant: (4)(1) − (3)(2) = 4 − 6 = −2.