Linear Algebra

50 Chapter One. Linear SystemsExerciseš 1.8 Use Gauss-Jordan reduction to solve each system.(a) x + y = 2 (b) x − z = 4 (c) 3x − 2y = 1x − y = 0 2x + 2y = 1 6x + y = 1/2(d) 2x − y = −1x + 3y − z = 5y + 2z = 5̌ 1.9 Find the reduced echelon ⎛ form of each ⎞ matrix. ⎛⎞( ) 1 3 11 0 3 1 22 1(a)(b) ⎝⎠ (c) ⎝⎠1 3(d)⎛0 1 3⎞2⎝0 0 5 6⎠1 5 1 52 0 4−1 −3 −31 4 2 1 53 4 8 1 2̌ 1.10 Find each solution set by using Gauss-Jordan reduction and then reading offthe parametrization.(a) 2x + y − z = 14x − y = 3(d) a + 2b + 3c + d − e = 13a − b + c + d + e = 3(b) x − z = 1y + 2z − w = 3x + 2y + 3z − w = 71.11 Give two distinct echelon form versions of this matrix.⎛⎞2 1 1 3⎝6 4 1 2⎠1 5 1 5̌ 1.12 List the reduced echelon forms possible for each size.(a) 2×2 (b) 2×3 (c) 3×2 (d) 3×3(c) x − y + z = 0y + w = 03x − 2y + 3z + w = 0−y − w = 0̌ 1.13 What results from applying Gauss-Jordan reduction to a nonsingular matrix?1.14 [Cleary] Consider the following relationship on the set of 2×2 matrices: we saythat A is sum-what like B if the sum of all of the entries in A is the same as thesum of all the entries in B. For instance, the zero matrix would be sum-what likethe matrix whose first row had two sevens, and whose second row had two negativesevens. Prove or disprove that this is an equivalence relation on the set of 2×2matrices.1.15 [Cleary] Consider the set of students in a class. Which of the following relationshipsare equivalence relations? Explain each answer in at least a sentence.(a) Two students x and y are related if x has taken at least as many math classesas y.(b) Students x and y are related if x and y have names that start with the sameletter.1.16 The proof of Lemma 1.5 contains a reference to the i ≠ j condition on the rowcombination operation.(a) The definition of row operations has an i ≠ j condition on the swap operationρ i ↔ ρ j . Show that in A ρ i↔ρ j−→ ρ i↔ρ j−→ A this condition is not needed.(b) Write down a 2×2 matrix with nonzero entries, and show that the −1 · ρ 1 + ρ 1operation is not reversed by 1 · ρ 1 + ρ 1 .