“main”2007/2/16page 139iiiiiiii2.4 Elementary Row Operations and Row-Echelon Matrices 1393. Verify that for all values of t,(1 − t,2 + 3t,3 − 2t)is a solution to the linear systemx1+ x2+ x3= 6,x1− x2− 2x3=−7,5x1+ x2− x3= 4.4. Verify that for all values of s and t,(s, s − 2t,2s + 3t,t)is a solution to the linear systemx1+ x2− x3+ 5x4= 0,2x2− x3+ 7x4= 0,4x1+ 2x2− 3x3+ 13x4= 0.5. By making a sketch in the xy-plane, prove that thefollowing linear system has no solution:2x + 3y = 1,2x + 3y = 2.For Problems 6–8, determine the coefficient matrix, A,theright-hand-side vector, b, and the augmented matrix A#ofthe given system.6.x1+ 2x2− 3x3= 1,2x1+ 4x2− 5x3= 2,7x1+ 2x2− x3= 3.7.x + y + z − w = 3,2x + 4y − 3z + 7w = 2.8.x1+ 2x2− x3= 0,2x1+ 3x2− 2x3= 0,5x1+ 6x2− 5x3= 0.For Problems 9–10, write the system of equations with thegiven coefficient matrix and right-hand-side vector.9. A =1 −12311−263142, b =1−12.10. A =2134 −12763, b =31−5.11. Consider the m × n homogeneous system of linearequationsAx = 0. (2.3.2)(a) If x =[x1x2... xn]Tand y =[y1y2... yn]Tare solutions to (2.3.2), show thatz = x + y and w = cxare also solutions, where c is an arbitrary scalar.(b) Is the result of (a) true when x and y are solu-tions to the nonhomogeneous system Ax = b?Explain.For Problems 12–15, write the vector formulation for thegiven system of differential equations.12. x1=−4x1+ 3x2+ 4t, x2= 6x1− 4x2+ t2.13. x1= t2x1− tx2, x2= (−sin t)x1+ x2.14. x1= e2tx2, x2+ (sin t)x1= 1.15. x1= (−sin t)x2+ x3+ t , x2=−etx1+ t2x3+ t3,x3=−tx1+ t2x2+ 1.For Problems 16–17 verify that the given vector function xdefines a solution to x= Ax + b for the given A and b.16. x(t) =e4t−2e4t,A=2 −1−23, b(t) =00.17. x(t) =4e−2t+ 2 sin t3e−2t− cos t,A=1 −4−32,b(t) =−2(cos t +sin t)7sint + 2 cos t.2.4 Elementary Row Operations and Row-Echelon MatricesIn the next section we will develop methods for solving a system of linear equations.These methods will consist of reducing a given system of equations to a new system thathas the same solution set as the given system but is easier to solve. In this section weintroduce the requisite mathematical results.“main”2007/2/16page 140iiiiiiii140 CHAPTER 2 Matrices and Systems of Linear EquationsElementary Row OperationsThe first step in deriving systematic procedures for solving a linear system is to determinewhat operations can be performed on such a system without altering its solution set.Example 2.4.1 Consider the system of equationsx1+ 2x2+ 4x3= 2, (2.4.1)2x1− 5x2+ 3x3= 6, (2.4.2)4x1+ 6x2− 7x3= 8. (2.4.3)Solution: If we permute (i.e., interchange), say, Equations (2.4.1) and (2.4.2), theresulting system is2x1− 5x2+ 3x3= 6,x1+ 2x2+ 4x3= 2,4x1+ 6x2− 7x3= 8,which certainly has the same solution set as the original system. Returning to the originalsystem, if we multiply, say, Equation (2.4.2) by 5, we obtain the systemx1+ 2x2+ 4x3= 2,10x1− 25x2+ 15x3= 30,4x1+ 6x2− 7x3= 8,which again has the same solution set as the original system. Finally, if we add, say,twice Equation (2.4.1) to Equation (2.4.3), we obtain the systemx1+ 2x2+ 4x3= 2, (2.4.4)2x1− 5x2+ 3x3= 6, (2.4.5)(4x1+ 6x2− 7x3) + 2(x1+ 2x2+ 4x3) = 8 + 2(2). (2.4.6)We can verify that, if (2.4.4)–(2.4.6) are satisfied, then so are (2.4.1)–(2.4.3), andvice versa. It follows that the system of equations (2.4.4)–(2.4.6) has the same solutionset as the original system of equations (2.4.1)–(2.4.3). More generally, similar reasoning can be used to show that the following threeoperations can be performed on any m × n system of linear equations without alteringthe solution set:1. Permute equations.2. Multiply an equation by a nonzero constant.3. Add a multiple of one equation to another equation.Since these operations involve changes only in the system coefficients and constants(and not changes in the variables), they can be represented by the following operationson the rows of the augmented matrix of the system:1. Permute rows.2. Multiply a row by a nonzero constant.3. Add a multiple of one row to another row.“main”2007/2/16page 141iiiiiiii2.4 Elementary Row Operations and Row-Echelon Matrices 141These three operations, called elementary row operations, will be a basic computationaltool throughout the text, even in cases when the matrix under consideration is not derivedfrom a system of linear equations. The following notation will be used to describeelementary row operations performed on a matrix A.1. Pij: Permute the ith and j th rows in A.2. Mi(k): Multiply every element of the ith row of A by a nonzero scalar k.3. Aij(k): Add to the elements of the j th row of A the scalar k times the correspondingelements of the ith row of A.Furthermore, the notation A ∼ B will mean that matrix B has been obtained frommatrix A by a sequence of elementary row operations. To reference a particular elemen-tary row operation used in, say, the nth step of the sequence of elementary row operations,we will writen∼ B.Example 2.4.2 The one-step operations performed on the system in Example 2.4.1 can be described asfollows using elementary row operations on the augmented matrix of the system:12422 −53646−781∼2 −536124246−781. P12. Permute (2.4.1) and (2.4.2).12422 −53646−781∼124210 −25 15 3046−781. M2(5). Multiply (2.4.2) by 5.12422 −53646−781∼12422 −53 66101121. A13(2). Add 2 times (2.4.1) to (2.4.3). It is important to realize that each elementary row operation is reversible; we can “un-do” a given elementary row operation by another elementary row operation to bring themodified linear system back into its original form. Specifically, in terms of the notationintroduced above, the reverse operations are determined as follows (ERO refers here to“elementary row operation”):ERO Applied to A Reverse ERO Applied to BA ∼ BB∼ APijPji: Permute row j and i in B.Mi(k)Mi(1/k): Multiply the ith row of B by 1/k.Aij(k)Aij(−k): Add to the elements of the jth rowof B the scalar −k times the correspondingelements of the ith row of BWe introduce a special term for matrices that are related via elementary row operations.DEFINITION 2.4.3Let A be an m × n matrix. Any matrix obtained from A by a finite sequence ofelementary row