NJ Topical Revision - Linear Algebra

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 72 of 103 F07: Matrices and Linear Spaces 1 (a) A square matrix A is called orthogonal if and .T T AA I A A I Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix. [2] (b) Let 1u , 2u , … , 1ku , ku be vectors in a vector space V such that  1 1,span , k V u u . Show that 1u , 2u , … , 1ku , ku are linearly dependent. [2] (c) (i) Let A be a square matrix and  is an eigenvalue of A with corresponding eigenvector u . Show that 2 is an ei genvalue of 2A with correspon ding eigenvector u . [2] (ii) Suppose further that 2 A A . Show that if  is an eigenvalue of A , then 0  or 1. [2] (2016 NYJC / JC1 / MYE / Q5) 2 It is given that the eigenvalues 1 , 2 , 3 of the matrix 3 1 1 8 8 8 1 1 1 4 2 4 1 1 5 8 8 8 a a a                   M , are the roots of the equation 3 2) )32( 48( 2 0.(2 ) 3a a a        Find 1 , 2 , 3 in terms of a. [3] Find matrices Q and D such that 1M QDQ , where the elements of Q are independent of a, and D is a diagonal matrix. [The evaluation of 1Q is not required. [5] Find the set of values of a such that all the elements of nM tend to zero as n . [3] (2016 NYJC / JC1 / MYE / Q6)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 73 of 103 3 A linear transformation 3 3:L   is defined by 2 , x x z L y x y z y az                    where a is a real constant. (a) Determine the conditions for a and b such that the point (2, 3, ) b is in the range of L . [3] (b) Find the value of a such that the null space of L is non-trivial. Find a basis for the range of L and a basis for the null space of L for this value of a. [5] For this value of a, (i) show that (0, 0, 0) is the only invariant point under L , [2] (ii) find, in the form  r a b , an equation for the line whose image under L is the point with position vector 5 3 2         , [3] (iii) show that the image under L of the plane 2x y z   is the plane 0x y z   . [3] (2016 NYJC / JC1 / MYE / Q7) 4 Let nP be the set of polynomials with real coefficients of the form 2 0 1 2( ) n np x a a x a x a x      . (i) Assume that nP is a vector space over  , write down the standard basis for thi s vector space. [1] (ii) Let { ( ) : (0) 0}nV p x P p   . Show that V is a subspace of nP . [3] Every polynomial with real coefficients of the form 2 0 1 2 n na a x a x a x     can be expressed as a ( 1) 1n  column vector 0 1 n a a a              . The I -transformation, which acts on elements of nP , is defined as 0 ( ) ( ) x I p p t dt . (iii) Show that 1: n nI P P  is a linear transformation. Write down the matrix representing I and find a basis for the range space of I . [5] (2016 NYJC / JC1 / Promo / Q4)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 74 of 103 5 (i) Find the eigenvalues and eigenvectors 1e , 2e , 3e of the matrix 4 2 0 2 3 2 0 2 2           A . [4] (ii) Show that the eigenvectors are mutually perpendicular. [2] (iii) Let 6 3 0         b . By taking scalar products with each of the eigenvec tors, find 1p , 2p and 3p such that 1 1 2 2 3 3p p p b e e e . [3] (iv) By writing 1 1 2 2 3 3q q q x e e e and using the decomposition in (iii), solve the equation A x b . [3] (2016 NYJC / JC1 / Promo / Q9) 6 Determine the values of a and b for which the linear system below has 2 7 7 2 3 17 2 2 ( 1) 3 x y z x y z a x y a z b           (a) no solution, (b) exactly one solution, (c) infinitely many solutions. Interpret the geometrical meaning of the system in part (a). [5] (2016 HCI / JC1 / Promo / Q3) 7 Let  33M  denote the linear space of all 3 3 real matrices under the usual addition and multiplication of matrices. Identify which of the following collection of 3 3 matrices form a subspace of  33M  . (a) All matrices of rank at most 1. (b) All non-invertible matrices. (c) Any matrix A satisfying 2 TA A O  . [8] (2016 HCI / JC1 / Promo / Q5) 8 The transformations 4 4 1T :   and 4 4 2T :   are represented by the matrices 1 1 2 3 2 0 1 1 1 0 0 1 1 0 0 0 0 M            and 2 1 1 2 0 2 0 3 1 1 1 0 2 3 1 1 3 M             respectively. The null space of 1T is denoted by 1K and the range space of 2T is denoted by 2R . (i) Find dim( 1K ) and a basis for 1K . [3] (ii) By using a graphing calculator, find dim( 2R ). [1] (iii) With the aid of the result in part (i), find a non-zero vector x in 4 such that 1 2M M x 0 . [4] (2016 HCI / JC1 / Promo / Q6)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 75 of 103 9 It is given that the eigenvalues of matrix 1 2 2 6 4 6 6 5 7 A          are 2, 1 and 1  with corresponding eigenvectors j k , i k and u respectively. Let the linear transformation 3 3L :   be    L Ax x . (i) Find u . [2] (ii) Find the Cartesian equation of the line through the origin, perpendicular to j and is invariant under L. [2] (iii) Find the set of points which are invariant under L. [2] (iv) Show that the plane 2 2 0x y z   is invariant under L. [3] (2016 HCI / JC1 / Promo / Q7) 10 Show that, for all real values of a , the rank of the matrix 1 2 3 1 3 3 1 1 1 3 1 2 3 6 2 1 a a a a a a a                    M , is equal to 2. [2] The null space of the linear transformation represented by M is denoted by K . The set 1 2{ , }e e is a basis for K , and the vectors 0x and b are such that 0 Mx b . (i) Show that if 0 1 2   x x e e , with  ,  , then Mx b . [2] (ii) Show that if Mx b , then 0( ) K x x , and deduce that x is of the form 0 1 2  x e e . [4] (iii) Find the vectors 1e and 2e which are of the forms 1 0 r s             and 0 1 t u             respectively, where r , s , t and u may depend on a . [3] (2017 NYJC / JC2 / BT / Q7) 11 Find the value of a for which the simultaneous equations 3 2 10, 5 4 17, 5 , x y z x y z x y az b          do not have a unique solution for x, y and z. [4] The equations represent three planes having a common line of intersection, L, find equations for L giving your answer in the form .x p y q z r l m n     [3] (2017 TJC / JC2 / BT / Q3)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 76 of 103 12 (i) A square matrix is said to be Markov if the sum of entries for each column is equal to 1. Show that the product of two 2 2 Markov matrices is also a Markov matrix. [2] (ii) Using (i), prove that if A is a 2 2 Markov matrix, then nA is a Markov matrix for all n   . [3] A Markov matrix is used when a situation can be modelled using Markov chain. Consider the following problem. Each year 100 % of the urban population of a country moves to the rural district while 100 % of the rural population moves to the u rban district where  and  are non-zero. Let na and nb be the urban and rural population respectively after n years. If we let n n n a b       x , the n 0 n n x A x where A is a 2 2 Markov matrix with 1 1          A and 0x represents the initial population. Denote ( ) ( ) 11 12 ( ) ( ) 21 22 n n n n n a a a a      A . (iii) Show that ( ) ( 1) 11 11 (1 )n na a        . Hence show that ( ) 11 (1 )n na            . [5] (iv) Show that A has two eigenvalues 1 and  where 1 . [3] (v) If x is an eigenvector of A corresponding to the eigenvalue 1, show that n A x x for a