A-LevelFurther MathematicsMatrices9231/12 · Oct/Nov 2013 · Q710 Marks

The square matrix A has λ as an eigenvalue with e as a corresponding eigenvector. Show that e is an eigenvector of A² and state the corresponding eigenvalue. Find the eigenvalues of the matrix B, where B = [[1, 3, 0], [2, 0, 2], [1, 1, 2]]. Find the eigenvalues of B⁴ + 2B² + 3I, where I is the 3 × 3 identity matrix.

(a)The square matrix A has λ as an eigenvalue with e as a corresponding eigenvector. Show that e is an eigenvector of A² and state the corresponding eigenvalue.[3]
(b)Find the eigenvalues of the matrix B, where B = [[1, 3, 0], [2, 0, 2], [1, 1, 2]].[4]
(c)Find the eigenvalues of B⁴ + 2B² + 3I, where I is the 3 × 3 identity matrix.[3]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

[B1] (a) Proves first result. Ae = λe
[M1A1] ⇒ A²e = AAe = A λe = λ Ae = λ²e ⇒ result. (e ≠ 0 ⇒ λ² is an eigenvalue of A².)
[M1A1] (b) Obtains eigenvalues of B. (1 – λ)(λ – 4)(λ + 2) = 0
[A1A1] ⇒ λ = −2, 1, 4
[M1A1] (c) Obtains eigenvalues of related matrix. 1⁴e + 2×1²e + 3e = 6e ⇒ 6 is an eigenvalue.
[A1] (−2)⁴e + 2×(−2)²e + 3e = 27e ⇒ 27 is an eigenvalue.
[A1] 4⁴e + 2×4²e + 3e = 291e ⇒ 291 is an eigenvalue.

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