32.x1+2x2+x4=4,3x2+x3+2x4=3,4x1-x3+2x4=2,x1+x3+3x4=1.2For each of the following systems, findhso that the system has solutions.33.x1+2x2+3x3=5,2x1-x2-2x3=2h,3x1-x2-x3=h.34.x1+x2+x3=2,2x1+3x2+2x3=5,x1+x2+(h2-5)x3=h.35.x3+2x4=1,-2x1+2x2-x3+4x4=h,x1-x2+x3-x4=1.36.x1+x2+x3+x4=1,x1+hx4=1,x2+hx3=1.37.For what value(s) ofhdoes the following system has solution for anyb1,b2,b3. For such value(s) ofh, solve the system by the method of reduction.x1+hx3+x4=b1,-x1+h2x3-(2 +h)x4=b2,x1+x2+2x4=b3.38.ConsiderA=0012-22-141-11-1,b=1h1.For whath, doesAx=bhas solution? Find the general solution of the homogeneous systemAx=O.39.Does the following homogeneous system has a nontrivial solution?40.Without performing any computation, justify that the system below has a nontrivial solution.x1-3x2+ 9x3-2x4+ 3x5=0,3x1+ 6x2+ 4x3-3x4-2x5=0,-x1+ 2x2-7x3+ 5x4+ 7x5=0,x1-5x2+ 4x3-3x4-8x5=0.132
6.4 Rank6.4Rank4Definition (Rank)Therankof a matrixA, writtenrankA, is equal to the number of pivots in a rowechelon form ofA.¥Example 6.4.1 (Rank)In Examples 6.2.1, 6.2.2, 6.2.3 and 6.2.4(page 121), we saw that the following augmented matrices•2-1-1-112‚,111123213822,111213-1401-12,1-133311501-2-1,which have the following row echelon forms"/£¡¢-1120/£¡¢13#,/£¡¢11110/£¡¢10-100/£¡¢-14,/£¡¢11120/£¡¢1-11000/£¡¢1,/£¡¢1-1330/£¡¢1-2-10000.The ranks of the four augmented matrices are 2, 3, 3, 2, respectively.2A systemAx=bof linear equations may contain many equations. However, after Gaussian elimination,some equations become the trivial equation 0 = 0. Therefore the rank of£Ab/is the “essential number”of equations, or the “true size” of the system.¥Example 6.4.2 (Rank)There seems to be many equations in the following systemx1-x2=1,2x1-2x2=2,-x1+x2=-1,5x1-5x2=5.However, all the equations can be deduced from the first one. There is essentially only one equation (i.e.,x1-x2= 1) in the system. Correspondingly, the rank of the matrix1-112-22-11-15-55is 1. In fact, one may verify this by finding its row echelon form1-112-22-11-15-55-2R1+R2, R1+R3------------→-5R1+R4/£¡¢1-11000000000.2Rank may be used to describe the existence and uniqueness for solutions of a systemAx=b.Notethat if we delete the last column of the row echelon form of£Ab/, then we get the row echelon form ofA. Therefore by the discussion of Section 6.3.3(page 128), we haverank£Ab/=(rankA,ifAx=bhas solutions,rankA+ 1,ifAx=bhas no solution.