Matrix and Linear Systems Questions

The TMA covers only chapters 1 and 2. It consists of four questions; you should give the details of your solutions and not just the final results. Q−1: [3×2 marks] answer each of the following as True or False justifying your answers: a) Every 𝑛 × 𝑛 matrix 𝐴 is the sum of a symmetric and a skew-symmetric matrix. b) Let D be a 3 × 3 diagonal matrix and 𝐴 be an arbitrary 3 × 3 matrix, the AD=DA. 1 𝑎2 0 c) The vectors [ 0 ] , [1] and [0] are linearly independent for all values of 𝑎. 2 1 1 MT132 / TMA Page 1 of 3 2022/2023 Summer Q−2: [6+2 marks] a) Find all values of 𝑎 for which the linear system −2𝑥 − 3𝑦 − 𝑧 = 0 −𝑥 − 𝑦 = −1 { 2 −𝑦 + (𝑎 − 1)𝑧 = 𝑎 + 1 Has: i. No solution; ii. a unique solution or iii. Has infinitely many solutions. 2𝑥 − 4𝑦 = 4 b) Solve the linear system { −2𝑥 + 3𝑦 = 3 MT132 / TMA Page 2 of 3 2022/2023 Summer Q−3: [(4+2) +2 marks] 1 0 1 1 a) Let 𝐴 = [1 −1 1] and 𝐵 = [0]. 5 2 1 0 i. Find, if it exists, 𝐴−1 . ii. Find the matrix 𝑋 such that 𝐴𝑋 = 𝐵 b) If the following system of equations has more than one solution, Find the value of a. 𝑎𝑥1 − 2𝑥2 − 𝑥3 = 0 (𝑎 + 1)𝑥2 + 4𝑥3 = 0 (𝑎 − 1)𝑥3 = 0 Q−4: [2+6 marks] 1 0 1 a) Let 𝑆 = {[0] , [ 1 ]} be a set of vectors in ℝ3 and 𝑋 = [−2] be a vector in ℝ3 . 1 −1 −2 If possible, write 𝑋 as a linear combination of vectors in 𝑆. b) Decide whether the given set of vectors is linearly independent or linearly dependent in the given vector space. 1 0 1 0 1 1 i. 1 , 0 , 1 𝑖𝑛 𝑅5 0 1 1 {[0] [1] [1]} 1 −1 0 ii. {[1] , [ 1 ] , [2]} 𝑖𝑛 𝑅3 0 1 1 1 2 4 iii. The row of the matrix 𝐴 = [ 0 3 5] as vector in 𝑅3 −1 1 1 MT132 / TMA Page 3 of 3 2022/2023 Summer