Preparation problems for the discussion sections on September 2nd and 4th1. For the following systems determine(1) the augmented matrix,(2) an echelon form of the matrix,(3) the reduced echelon form of the matrix,(4) whether the system is consistent,(5) the set of solutions (in parametric form),(6) how many solutions the system has,(7) the geometric interpretation of the set of solutions.System A:x2= 3x1+ 2x2= 4System B:x1+ x2= 32x1+ 2x2= 6System C:x1+ x2= 32x1+ 2x2= 72. Some questions to check your understanding:a) What is the largest possible number of pivots a 4 × 6 matrix can have? Why?b) What is the largest possible number of pivots a 6 × 4 matrix can have? Why?c) How many solutions does a consistent linear system of 3 equations and 4 unknownshave? Why?d) Suppose the coefficient matrix corresponding to a linear system is 4 × 6 and has 3 pivotcolumns. How many pivot columns does the augmented matrix have if the linear systemis inconsistent?3. Find a parametric description of the set of solutions of:x1+ 3x2− 5x3= 4x1+ 4x2− 8x3= 7−3x1− 7x2+ 9x3= −64. For which values of h1and h2is the following system consistent?x1= h1x2= 5x1+ 2x2= h25. Show that the interchange of two rows of a matrix can be accomplished by a finite sequenceof elementary row operations of the other two types.16. Let A = [aij]3×4, and let B = [bij]3×4be an echelon form of A.(1) Is it true that, if a11= 0, then b11= 0?(2) Is it true that, if A has a column of zeros, then B also has a column of zeros?(3) Suppose B has a row of zeros. What can you say about rows of A? (Explain.)(4) Suppose we form a new matrix using some columns of A, let’s say the first and the thirdcolumn. What is an echelon form corresponding to this new
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