Math 1b — Row-equivalence; matrix inversesJanuary 7, 2011Recall that matrices A and B are row-equivalent when one can be obtained from theother by a sequence of elementary ro w operations.An elementary row operation on a matrix M givesusamatrixwhoserowsMwhoserows are linear combinations of the rows of M. Since elementary row operations can be‘undone’ by other elementary row operations, the rows of M are linear combinations of therows of M. It follows that if A and B are row-equivalent, then the rows of A ar e linearcombinations of the rows of B,andtherowsofB are linear combinations of the rows ofA. (Later, we will say that A and B have the s ame row space.)To review one item from the handout on matrix multiplication, recall that givenmatrices A and B, the equation A = CB holds for some matrix C if and only if the rowsof A are linear combinations of the rows of B. For example,a11a12a13a21a22a23 =2 −34567−89 ⎛⎜⎝b11b12b13b21b22b23b31b32b33b41b42b43⎞⎟⎠means that(a11,a12,a13)=2(b11,b12,b13) − 3(b21,b22,b23)+4(b31,b32,b33)+5(b41,b42,b43)and(a21,a22,a23)=6(b11,b12,b13)+7(b21,b22,b23) − 8(b31,b32,b33)+9(b41,b42,b43).So if M and N are row-equivalent, there are square matrices S and T so that M = SNand N = TM.InversesThe way I use the terms, a square matrix A is nonsingular when its echelon form isthe identity matrix I,andisinvertible when there exists a matrix B of the same size sothat AB = BA = I. Such a matrix is called the inverse of A and is denoted by A−1.Notethat if A is invertible, then Ax = b is equivalent to x = A−1b.For example, we may check that3524 −1=2 −5/2−13/2 .Theorem. A square matrix is invertible if and only if it is nonsingular.We prove only part of the theorem here, by explaining how to find A−1when A isrow-equivalent to I. Suppose A is row-equivalent to I (both n × n). Start with the n × 2nmatrixM =A Iand use row operations or p ivots to get toM=I Bfor some matrix B. We claim that for this B, AB = BA = I.Therearevariouswaystoexplainwhythisworks.Hereisaone.Since [A, I]and[I, B] are row-equivalent, there are square matrices S and T so that[A, I]=S[I, B]and[I, B]=T [A, I]. That is, [A, I]=[SI,SB]=[S, SB]and[I, B]=[TA,TI]=[TA,T]. The fir st equation m eans A = S and I = SB,soAB = I. The secondequation means I = TA and B = T ,soBA = I.  For another explanation via elementary matrices, see Section 2.4 of LAD W . (We willnot emphasize elementary matrices in this course.)Uniqueness of echelon formTheorem. Two matrices A and B are row-equivalent if and only if they have the sam ereduced echelon form.(I really shouldn’t say the reduced echelon form, until I am SURE that only one matrixin reduced echelon form is r ow-equivalent to a given matrix.)Proof: If A and B have the same reduced echelon form E,thenA is row-equivalent to Eand E is row-equivalent to B. It follows that A is row-equivalent to B.Now suppose A and B are row equivalent. Let E1be a reduced echelon form of Aand E2be a reduced echelon form of B.ThenE1and E2are row-equ ivalent. We wantto show E1= E2. A complete discussion requires more details than I want to type. Butsupp ose we know that the special columns of both E1and E2occur right away, at the leftof the matrices, and that neither has rows of all zer os . (This is a BIG assumption.) Thatis, supposeE1=IF1and E2=IF2.Since E1and E2are row equivalent, E2= CE1for some matrix C. This means I = CIand F2= CF1.ButthenC = I and F2= F1.