A-LevelFurther MathematicsMatrices9231/12 · May/June 2013 · Q69 Marks
The matrix A is given by A = [[4, -5, 3], [3, -4, 3], [1, -1, 2]] Show that e = [[1], [1], [1]] is an eigenvector of A and state the corresponding eigenvalue. Find the other two eigenvalues of A. The matrix B is given by B = [[-1, 4, 0], [-1, 3, 1], [1, -1, 3]] Show that e is an eigenvector of B and deduce an eigenvector of the matrix AB, stating the corresponding eigenvalue.
- (a)The matrix A is given by A = [[4, -5, 3], [3, -4, 3], [1, -1, 2]] Show that e = [[1], [1], [1]] is an eigenvector of A and state the corresponding eigenvalue.[2]
- (b)Find the other two eigenvalues of A.[4]
- (c)The matrix B is given by B = [[-1, 4, 0], [-1, 3, 1], [1, -1, 3]] Show that e is an eigenvector of B and deduce an eigenvector of the matrix AB, stating the corresponding eigenvalue.[3]
Mark scheme
Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.
- M1A1 Shows e is an eigenvector of A and gives eigenvalue. Ae = 2e ⇒ e is an eigenvector with eigenvalue 2.
- M1A1 Finds characteristic equation. λ³ – 2λ² – λ + 2 = 0
- A1 Factorises. ⇒ (λ – 2)(λ² – 1) = 0
- A1 States other eigenvalues. Other eigenvalues are -1 and 1.
- B1 Repeats for B. Be = 3e ⇒ e is an eigenvector with eigenvalue 3
- M1A1 States result for AB. ABe = A.3e = 3Ae = 3.2e = 6e. AB has eigenvector e with eigenvalue 6
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