A right inverse of A is a solution X of the matrix division problem AX = I. When matrices are allowed to have rational entries, many matrices that have no right inverses with integer entries do now have right inverses.
For example, matrix division gives infinitely many solutions of the equation AX = I, where A is the 2x3 matrix whose first row is 1, 2, 3 and whose second row is 4, 5, 6, and where I is the 3x3 identity matrix. Since A has rank 2 (for example, the 2x2 minor that leaves out the last column is not zero), a diagonal matrix equivalent to D has no nonzero row, and the most general solution Z of DZ = M has as its first two rows the first two rows of M divided by the corresponding entries of D and its last row is arbitrary. The D given by quick(A) has diagonal entries 1 and 3. The solution Z of DZ = M in which the last row is zero can be entered, once M has been found, by giving the commands Z = zeros(3,2), Z(1,1:2) = M(1,1:2), Z(2,1:2) = M(2,1:2)/3. Other solutions Z can be found by changing the entries in the last row, for example, by giving the command Z(3,1:2) = [5 10]. For any matrix Z obtained in this way, X = N*Z will be a right inverse of A.
Note that there are infinitely many right inverses of this matrix A, but that each of them has at least one entry that is not an integer.
Another example you might try is the 2x3 matrix A whose first row is 1, 1/2, 1/3 and whose second row is 1/4, 1/5, 1/6. Since 60 is a common denominator of the fractions in A, give the command B = round(60*A) to find a matrix with integer entries whose diagonalization gives a diagonalization of A. Specifically, the command [E, M, N] = quick(B) will give unimodular matrices M and N such that MAN is the diagonal matrix E/60. You will then see that the commands Z = zeros(3,2), Z(1,1:2) = 60*M(1,1:2), Z(2,1:2) = 2*m(2,1:2), X = N*Z give a right inverse X of A. The most general right inverse X of A is obtained from this one by adding a multiple of the last column of N to each column.
Use this method to find right inverses of other 2x3 matrices with rational entries. What condition must a 2x3 matrix satisfy in order for there to be a right inverse? Does such a matrix ever have a left inverse?
The 3x3 matrix whose rows are 1, 2, 3 and 4, 5, 6 and 7, 8, 0 has a unique right inverse. Use matrix division to find it. (You can also find it by entering the matrix as A and giving the command inv(A).) Is it a left inverse?
Try other 3x3 examples. When does a right inverse exist? What is the role of the determinant? Find right inverses in a few cases. When is a right inverse of a 3x3 matrix also a left inverse? Reread Section 3 with these examples in mind.
Return to Chapter 6.