Question

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The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system 1-1 0 0 5 0 1-3 0-4 o 0 1-2 3 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice OA The solution set contains one solution: O B. The solution set is empty. Type integers or simplified fractions.) C. The solution has infinitely many elements

Answer 1

Given The Augmented matrix [1 0 0 0 -1 1 0 0 0 -3 1 0 0 0 -2 1 ellipsis -5 -4 3 4] Let AB = [1 0 0 0 -1 1 0 0 0 -3 1 0 0 0 -2 1 ellipsis -5 -4 3 4] Rank[A] = 4 (number of non-zero rows) and Rank[AB] = 4 therefore n = 4 (number of unknows) therefore Rank[A] = Rank[AB] = n Hence the given system having unique solution (one solution) By back substitution x_4 = 4, x_3 -2x_4 = 3, x_2 -3x_3 = -4, x_1 - x_2 = -5 x_3 = 3 + 2x_4, x_3 = 3 + 2(4) = 3 + 8 x_3 = 11 rightdoublearrow x_2 = -4 + 3x_3, rightdoublearrow x_2 = -4 + 3(11) = -4 + 33 x_2 = 29 rightdoublearrow x_1 = -5 + x_2 = -5 + 29 x_1 = 24