21 22 CHAPTER 1 Linear Equations in Linear Algebra3.9.12. a. 0011.13.14.C.d.15. a.b.1461000-9-6110030000301-200Го257los BaodisabRow reduce the matrices in Exercises 3 and 4 to reduced echelonform. Circle the pivot positions in the final matrix and in theoriginal matrix, and list the pivot columns.ETRY4.10010 01-30100011010003 49 700120000 01 011061 -6 57-63 46 77 8 9 Instavin05. Describe the possible echelon forms of a nonzero 2 x 2matrix. Use the symbols, *, and 0, as in the first part ofExample 1.6. Repeat Exercise 5 for a nonzero 3 x 2 matrix.Find the general solutions of the systems whose augmented ma-trices are given in Exercises 7-14.7.-4 2 012 -618 -4 01012 20 30 0]2 -5 -61 -6-30 0 010****■0 ■ 0*000 000*b.0 -1 0 -2700-4 101 9 40 0 00-50 200 0 0 08.10.*12.100HAS[13355 7[2[31101gas zoldsha00100-11471 -2 -1-6 -20015 7799 1Exercises 15 and 16 use the notation of Example 1 for matricesin echelon form. Suppose each matrix represents the augmentedmatrix for a system of linear equations. In each case, determine ifthe system is consistent. If the system is consistent, determine ifthe solution is unique.7110072]0 6-750 1-2-2 -37-42716. a.OWowb.17.20■0403In Exercises 17 and 18, determine the value(s) of h such that thematrix is the augmented matrix of a consistent linear system.h47]6:)0 0Jinsi In Exercises 19 and 20, choose h and k such that the system has(a) no solution, (b) a unique solution, and (c) many solutions. Giveseparate answers for each part.00019. x₁ + hx₂ = 2Banom 4x₁ + 8x₂ = k*■3*18.1 -3 -25h -720. x₁ + 3x₂ = 23x₁ + hx₂ = kb. The row reduction algorithm applies only to augmentedis toxins in matrices for a linear system.In Exercises 21 and 22, mark each statement True or False. Justifyeach answer.421//a.In some cases, a matrix may be row reduced to morethan one matrix in reduced echelon form, using differentsequences of row operations.c. A basic variable in a linear system is a variable thatcorresponds to a pivot column in the coefficient matrix.d. Finding a parametric description of the solution set of a22400 linear system is the same as solving the system.lulee. If one row in an echelon form of an augmented matrixis [0 0 0 5 0], then the associated linear system isinconsistent.22. a The echelon form of a matrix is unique.2b. The pivot positions in a matrix depend on whether rowinterchanges are used in the row reduction process.c. Reducing a matrix to echelon form is called the forwardphase of the row reduction process.d. Whenever a system has free variables, the solution setcontains many solutions.e. A general solution of a system is an explicit descriptionof all solutions of the system.23. Suppose a 3 x 5 coefficient matrix for a system has threepivot columns. Is the system consistent? Why or why not?24. Suppose a system of linear equations has a 3 x 5 augmentedmatrix whose fifth column is a pivot column. Is the systemconsistent? Why (or why not)?4 True/false questions of this type will appear in many sections. Methodsfor justifying your answers were described before Exercises 23 and 24 inSection 1.1.

Introduction A matrix is in reduced row-echelon form if it satisfies the following conditions: 1)…

21/a. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. b. The row reduction algorithm applies only to augmented matrices for a linear system. c. A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. d. Finding a parametric description of the solution set of a linear system is the same as solving the system. Mala e. If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.