A-LevelFurther MathematicsMatrices9231/12 · Oct/Nov 2016 · Q37 Marks
Find a matrix A whose eigenvalues are -1, 1, 2 and for which corresponding eigenvectors are (1) (0), (0) (1) (1), (1) (0) (1) (1) respectively.
- Find a matrix A whose eigenvalues are -1, 1, 2 and for which corresponding eigenvectors are (1) (0), (0) (1) (1), (1) (0) (1) (1) respectively.[7]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.
- [B1B1] P=(1 1 0) (0 1 1) (0 0 1) , D=(-1 0 0) (0 1 0) soi (0 0 2)
- [M1] P⁻¹AP = D ⇒ A = PDP⁻¹ soi
- [M1A1] P⁻¹ = (1 -1 1) (0 1 -1) (0 0 1)
- [M1] PD = (-1 1 0) ( 0 1 2) ( 0 0 2) or DP⁻¹ = (-1 1 -1) ( 0 1 -1) ( 0 0 2)
- [A1] A = (-1 2 -2) ( 0 1 1) ( 0 0 2)
- ALT METHOD: [M1, A1] Av₁ = -v₁, Av₂ = v₂, Av₃ = 2v₃
- [M1, A1, A1, A1] Multiply out, solve each equation
- [B1√] A = (-1 2 -2) ( 0 1 1) ( 0 0 2)
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