Linear Algebra

Section III. Reduced Echelon Form 51The diagram below shows the collection of all matrices as a box. Inside thatbox, each matrix lies in some class. Matrices are in the same class if and only ifthey are interreducible. The classes are disjoint — no matrix is in two distinctclasses. The collection of matrices has been partitioned into row equivalenceclasses. ∗ AB . . .One of the classes in this partition is the cluster of matrices shown above,expanded to include all of the nonsingular 2×2 matrices.The next subsection proves that the reduced echelon form of a matrix isunique; that every matrix reduces to one and only one reduced echelon formmatrix. Rephrased in terms of the row-equivalence relationship, we shall provethat every matrix is row equivalent to one and only one reduced echelon formmatrix. In terms of the partition what we shall prove is: every equivalenceclass contains one and only one reduced echelon form matrix. So each reducedechelon form matrix serves as a representative of its class.After that proof we shall, as mentioned in the introduction to this section,have a way to decide if one matrix can be derived from another by row reduction.We just apply the Gauss-Jordan procedure to both and see whether or not theycome to the same reduced echelon form.Exerciseš 1.7 Use Gauss-Jordan reduction to solve each system.(a) x + y = 2 (b) x − z = 4 (c) 3x − 2y = 1x − y = 0 2x + 2y = 1 6x + y = 1/2(d) 2x − y = −1x + 3y − z = 5y + 2z = 5̌ 1.8 Find the reduced echelon form of each matrix.( ) ( ) ( )1 3 11 0 3 1 22 1(a)(b) 2 0 4 (c) 1 4 2 1 51 3−1 −3 −33 4 8 1 2( )0 1 3 2(d) 0 0 5 61 5 1 5̌ 1.9 Find each solution set by using Gauss-Jordan reduction, then reading off theparametrization.(a) 2x + y − z = 14x − y = 3(d) a + 2b + 3c + d − e = 13a − b + c + d + e = 3(b) x − z = 1y + 2z − w = 3x + 2y + 3z − w = 7(c) x − y + z = 0y + w = 03x − 2y + 3z + w = 0−y − w = 0∗ More information on partitions and class representatives is in the appendix.