F07 - Matrices and Linear Spaces - Tutorial Set 1

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 1 of 13 National Junior College 2016 – 2017 H2 Further Mathematics Topic F7: Matrices and Linear Spaces (Tutorial Set 1) This tutorial set is for the following sections from the notes: §1 System of Linear Equations §2 Matrices and Matrix Operations §3 Inverse Matrix and Its Applications §4 Determinants Basic Mastery Questions 1 Without using a graphic calculator, find a row-echelon form and the reduced row-echelon form of the matrix 2 3 3 25 3 2 3 24 4 1 2 21         . 2 It is given that 3 2 0 1 6 a     A and 1 0 2 1 3 0 0 3 2          B . Find TAB and TBA in terms of a. 3 Which of the following are elementary matrices? State the inverses of these elementary matrices. 0 1 0 1 0 0 0 0 1        A , 2 1 1 0     B , 1 0 0 0 2 0 0 0 1          C , 1 0 0 6 1 0 0 0 1        D , 1 0 0 0 1 0 0 1 2        E 4 Find the determinant and the inverse of 2016 2017 2018 2019      . 5 Show that 1 1 1 a b c b c a c a b           is singular. www.KiasuExamPaper.com 638

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 2 of 13 6 Let 1 3 0 0 1 1 2 7 0         A . (i) Find  det A using (a) cofactor expansion; (b) the special rule that only works for 3 3 matrix; (c) row reductions (to a upper-triangular or diagonal matrix); (c) a graphic calculator. (ii) Find 1A using (a) elementary row operations; (b) its adjoint; (c) a graphic calculator. 7 Solve the following system of linear equations 2 3 3 25 3 2 3 24 4 2 21 x y z x y z x y z          using (a) Gaussian elimination method; (b) Gauss-Jordan elimination method; (c) the inverse of the coefficient matrix; (d) Cramer’s Rule, (e) a graphic calculator. What can you say about the relationship among the three planes with these equations? 8 Solve the following linear system 2 3 4 5 3 5 7 11 3 2 7 x y z w x y z w x z w            using (a) Gauss-Jordan elimination method; (b) a graphic calculator. 9 Express the matrix 2 3 1 0      as a product of elementary matrices. www.KiasuExamPaper.com 639

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 3 of 13 Practice Questions 10 Determine the value(s) of a such that 2 5 1 1 1 3 10 2 a a          is singular. Hence, discuss how the value of b affects the number of the solutions of the linear system, and give a geometrical interpretation of the solutions for each case. 2 5 0 2 3 10 2 10 x y z x by z x y bz           11 Use row reduction to show that    1 2 3 2 1 3 2 3 1 2 2 2 1 2 3 1 1 1 a a a a a a a a a a a a     . 12 (a) Given 0b , prove that 2 2 0 2 a a b a b a b a b a a b a a b        if and only if a b . (b) Prove that 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 b c c a a b a b c b c c a a b a b c b c c a a b a b c           . (1973 A Level / FM / Nov / P2) 13 Show that if n A O for some integer 1n , then A is not invertible. 14 Let A be the matrix a b c d e f g h i         . Show that if 0a b c d e f g h i         , then A is not invertible. 15 Show that there does not exist an n n matrix A, where n is odd, such that 2 A I O . 16 What can you say about a and b if the matrix 1 4 3 2 8 1 3 a b         A is singular? www.KiasuExamPaper.com 640

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 4 of 13 17 Let  ijdD be an n n diagonal matrix. (i) Show that mD is also an n n diagonal matrix with entries on the main diagonal, 11 nd , 22 nd , …, m nnd , for all positive integer m. (ii) It is further given 0iid  for all 1 i n  . Explain whether (i) holds if m is a negative integer instead. 18 Given matrices A and B where 2 1 3 1 0 4 3 1 0         A and 2 1 3 1 1 12 3 1 0         B , find non-singular matrices P and Q such that PA and QB are echelon matrices in which the first non-zero entry in any row is unity. For each of A and B find its inverse, if it exists. For each of (a) and (b) below, solve, if possible, the equation: (a) 4 1 5        Ax , (b) 4 1 5        Bx . (1985 A Level / FM / Nov / P2) 19 Given that A is an invertible 3 3 matrix, show that the first column of 1A is the solution of the equation 1 0 0        Ax . Give equations whose solutions are the second and third columns of 1A respectively. Hence or otherwise, find the inverse of P, where 1 0 1 0 0 1 a b c        P . Find the matrix B given that 3 1 2 1 0 5     BP . (1973 A Level / FM / Jun / P1) www.KiasuExamPaper.com 641

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 5 of 13 20 (i) By considering 1 2 1 1 a a and 1 2 3 2 2 2 1 2 3 1 1 1 a a a a a a (from PQ 11), form a conjecture for 1 2 3 4 2 2 2 2 1 2 3 4 3 3 3 3 1 2 3 4 1 1 1 1 a a a a a a a a a a a a . (ii) Using factor theorem or otherwise, prove your conjecture. (iii) Generalise and prove the result for the n n matrix 1 2 1 1 1 1 2 1 1 1 n n n n n a a a a a a         . 21 (a) The matrices A and B are such that AB BA . Show that   3 3 2 2 33 3    A B A A B AB B . (b) Let C be a given matrix where C O . Find two different matrices P satisfying both the equations CP PC and 2 26  P PC C O . (c) Given that 1 1 1 0     D , show that all the solutions of the equation QD DQ are of the form   Q I D where  and  are scalars. (1982 A Level / FM / Jun / P2) 22 A square matrix A is said to be symmetric if T A A and skew-symmetric if T A A . For each of the following statements, either prove it, or give a counterexample to show that it is false. (i) A is a 2 2 non-singular symmetric matrix  1A is symmetric. (ii) A is a 2 2 skew-symmetric matrix   det 0 A . (iii) A is a 3 3 skew-symmetric matrix   det 0 A . (iv) TA A is a skew-symmetric matrix, where A is any 2 2 matrix. (v) Any 2 2 matrix can be expressed as the sum of a symmetric an a skew-symmetric matrix. (1977 A Level / FM / Jun / P2) www.KiasuExamPaper.com 642

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Matrices and Linear Spaces (Tutorial Set 1) Page 6 of 13 23 Show that the matrix 1 2 4 3     A satisfies the equation 2 4 5  A A I O , where I and O denote the 2 2 identity and zero matrices respectively. Prove by induction that, for each positive integer n, there are real numbers nb and nc such that n n nb c A A I . Hence or otherwise, find the matrix B without using a calculator, where 4 3 23 7 10 6    B A A A A I . (1976 A Level / FM / Jun / P2) 24 (i) Let A be the matrix 1 2 3 4 1 2 3 4 1 2 3 4 a a a a b b b b c c c c         . Write down matrices 1P and 2P such that 1 2 3 4 1 1 1 2 2 3 3 4 4 1 2 3 4 a a a a b a b a b a b a c c c c            P A ; 1 2 3 4 2 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 a a a a b a b a b a b a c ka c ka c ka c ka               P A . (ii) Given that 1 3 2 4 2 1 3 6 4 9 13 25             B , find a matrix P such that 1 3 2 4 0 1 0 0 x y s t        PB . Solve the equation 5 3 1        Bx . (1982 A Level / FM / Nov / P1) www.KiasuExamPaper.com 643

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