1. 0 2 . Consider the matrix A = a 1 (a) Find the matrix A2. (2) (b) If det A2 = 16, determine the possible values of a. (3) (Total 5 marks) 2. Consider the matrices 3 2 1 3 , B . A 5 4 2 2 (a) Find BA. (2) (b) Calculate det (BA). (2) (c) Find A(A–1B + 2A–1)A. (3) (Total 7 marks) 3. Let A, B and C be non-singular 2×2 matrices, I the 2×2 identity matrix and k a scalar. The following statements are incorrect. For each statement, write down the correct version of the right hand side. (a) (A + B)2 = A2 + 2AB + B2 (2) (b) (A – kI)3 = A3 – 3kA2 + 3k2A – k3 (2) IB Questionbank Mathematics Higher Level 3rd edition 1 (c) CA = B C = B A (2) (Total 6 marks) 4. (a) Write down the inverse of the matrix 1 3 1 A = 2 2 1 1 5 3 (2) (b) Hence, find the point of intersection of the three planes. x – 3y + z = 1 2x + 2y – z = 2 x – 5y + 3z = 3 (3) (c) A fourth plane with equation x + y + z = d passes through the point of intersection. Find the value of d. (1) (Total 6 marks) 5. ex Consider the matrix A = x 2 e e x , where x 1 . Find the value of x for which A is singular. (Total 6 marks) 6. The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I – X) is I + X + X2. (Total 6 marks) 7. Matrices A, B and C are defined as IB Questionbank Mathematics Higher Level 3rd edition 2 5 1 1 1 2 1 8 A 3 1 3 , B 3 1 0 , C 0 . 9 3 7 0 3 1 4 (a) a 0 0 Given that AB = 0 a 0 , find a. 0 0 a (1) (b) Hence, or otherwise, find A–1. (2) (c) Find the matrix X, such that AX = C. (2) (Total 5 marks) 8. 2 1 and that M2 – 6M + kI = 0 find k. Given that M = 3 4 (Total 5 marks) IB Questionbank Mathematics Higher Level 3rd edition 3