Matrix and Linear Systems Questions
August 1, 2024
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. Qβ1: [3Γ2 marks] answer each of the following as True or False justifying your answers: a) Every π Γ π matrix π΄ is the sum of a symmetric and a skew-symmetric matrix. b) Let D be a 3 Γ 3 diagonal matrix and π΄ be an arbitrary 3 Γ 3 matrix, the AD=DA. 1 π2 0 c) The vectors [ 0 ] , [1] and [0] are linearly independent for all values of π. 2 1 1 MT132 / TMA Page 1 of 3 2022/2023 Summer Qβ2: [6 2 marks] a) Find all values of π for which the linear system β2π₯ β 3π¦ β π§ = 0 βπ₯ β π¦ = β1 { 2 βπ¦ (π β 1)π§ = π 1 Has: i. No solution; ii. a unique solution or iii. Has infinitely many solutions. 2π₯ β 4π¦ = 4 b) Solve the linear system { β2π₯ 3π¦ = 3 MT132 / TMA Page 2 of 3 2022/2023 Summer Qβ3: [(4 2) 2 marks] 1 0 1 1 a) Let π΄ = [1 β1 1] and π΅ = [0]. 5 2 1 0 i. Find, if it exists, π΄β1 . ii. Find the matrix π such that π΄π = π΅ b) If the following system of equations has more than one solution, Find the value of a. ππ₯1 β 2π₯2 β π₯3 = 0 (π 1)π₯2 4π₯3 = 0 (π β 1)π₯3 = 0 Qβ4: [2 6 marks] 1 0 1 a) Let π = {[0] , [ 1 ]} be a set of vectors in β3 and π = [β2] be a vector in β3 . 1 β1 β2 If possible, write π as a linear combination of vectors in π. 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 ππ π 5 0 1 1 {[0] [1] [1]} 1 β1 0 ii. {[1] , [ 1 ] , [2]} ππ π 3 0 1 1 1 2 4 iii. The row of the matrix π΄ = [ 0 3 5] as vector in π 3 β1 1 1 MT132 / TMA Page 3 of 3 2022/2023 Summer
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