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Course 111: Algebra, 2nd Feb 2007 To be handed in at tutorials on Feb 5th and 6th. 1. Solve the following system of linear equations using both Gaussian elimination and Gauss-Jordan elimination. 3x1 − 4x2 = 10 −5x1 + 8x2 = −17 −3x1 + 12x2 = −12 What is the rank of the coefficient matrix describing this system? 2. Solve the following system of linear equations using both Gaussian elimination and Gauss-Jordan elimination. 7x1 + 2x2 − 2x3 − 4x4 + 3x5 = 8 −3x1 − 3x2 + 2x4 + x5 = −1 4x1 − x2 − 8x3 + 20x5 = 1 What is the rank of the coefficient matrix describing this system? 3. Prove that if A is an n × n matrix (ie. A describes a system with as many equations as unknowns) then the reduced row-echelon form of the matrix will either contain at least one row of all zeroes or it will be the n × n identity matrix, In where 1 ... . . I = .. . . 0 ... 0 .. . {z } | 1 n cols, n rows