NYJC 2026 FM TP - Linear Algebra Set 2

NANYANG JUNIOR COLLEGE DEPARTMENT OF MATHEMATICS 2026 J2 H2 FM Linear Algebra (Set 2) Instruction: Attempt the questions marked with *. 1* The matrices A and B are defined as follows. cos sin , with 0sin cos   −= A and 10 01 = − B . The transformations 1T and 2T from 2  to 2  are defined by 1T: xx yy    →       A and 2T : . xx yy    →       B (a) By considering cos sin xr yr      =       for 0r and  0, 2 , describe the geometrical transformation 1 T , explaining your answer. [2] (b) Describe 2T geometrically. [1] Let matrix M be 11 322 .11 322  −   (c) Without using a calculator, state the smallest positive integer value of k such that kM gives the identity matrix and explain your answer. [2] (d) Given that 1 =, − C M B M find the Cartesian equations of the two lines through the origin which are invariant under the transformation given by the matrix C. [3] 2* (a) The matrix A has eigenvectors 1 2 1  − , 2 0 1    , and 0 1 0    with corresponding eigenvalues 1− , 3 and 1 respectively. Find A. [2] (b) The vectors 1 2 3,,y y y are defined as: 1 7 4 5  =− y , 2 1 8 13 − = y , 3 66 60 6  =− y . Given that Q is a 3 x 3 matrix, determine whether or not the vectors 1 2 3,,Qy Qy Qy are linearly independent, justifying your conclusion. [2]

(c) (i) Show that the eigenvalues of the matrix 1 2 43 21 65 a a aa − =−− A are independent of a, where a is a non-zero constant. [3] (ii) Obtain, in terms of a, an eigenvector corresponding to each eigenvalue. [3] (iii) Hence find a matrix P and a diagonal matrix D such that 1 3 1( 2 )−−+=A I PDP . [2] 3 Let 32:T →  be a linear transformation defined by 2x xyTy x y zz  + = −+ for all 3 x y z     . (a) Find a basis for the range of T and explain why the dimension of the kernel of T is 1. [3] (b) Find a basis 1v for the kernel of T, where 1v is in the form 1 1 6 x y    and 11, xy   are constants to be determined. Hence, find a general solution 2 9 x Ty z   =  . [3] (c) Let 2 2 3 32 a ab a  = − + − v and 2 2 3 2 9 29 a a a  = − − −− v . Determine the conditions , ab   such that   3 1 2 3span , , =v v v  , explaining your answers clearly. [4]