Linear Algebra: Matrix Algebra and Determinants

Please show all your work clearly! Thank you!

(1) Consider the following augmented matrix in which * denotes an arbitrary real (or complex) number and I denotes a nonzero real (or complex) number. Determine whether the given augmented matrix corresponds to a consistent system. If so, is the solution unique? * 0 0 0 * (2) Same question as in exercise 1 but with the augmented matrix ** * * O* * O* O 0 0 0 * (3) If a linear system has more equations that variables, must it have a solution? If so, give an example and if not, explain why not. (4) Choose h and k so that the augmented matrix ( 2 " 2 ) has: (a) a unique solution (verify your assertion). (b) no solution (verify your assertion). (c) infinitely many solutions (verify your assertion). (5) Find all solutions to the system whose augmented matrix is 2 HOO 2 2 2 You must row-reduce the matrix "by hand." You must also carefully recored, using the notation introduced in class, every row operation carried out. Clearly identify basic and free variables