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