Let be a field, let , and let denote the multiplicative semigroup of matrices over . Characterize the units and zero divisors in . Does have a kernel?
In this previous exercise, we showed that a matrix is invertible if and only if it is row equivalent to the identity matrix. In this previous exercise, we showed that a matrix is row equivalent to the identity if and only if it has nonzero determinant (remember, this is over a field). So a matrix is invertible if and only if it has nonzero determinant.
Now suppose a matrix has determinant 0. Then as a linear transformation, is not injective. So there exists a nonzero column matrix such that . Then , and in particular is a zero divisor.
That is, every matrix is either a unit or a zero divisor, and to distinguish the two cases, we need only look at the determinant.
Note that has a zero, namely the zero matrix. Now every ideal in contains zero, so that the kernel (i.e. the intersection of all two-sided ideals) is nonempty.
Suppose is an ideal. Note that if contains a unit , then for all we have , so that . So every nontrivial ideal of is contained in the set of zero divisors.
Let be an ideal, and suppose there exists a nonzero matrix . Suppose the entry of is nonzero. By left- and right- multiplying by appropriate permutation matrices, we can construct matrices whose entry is nonzero for all . By this previous exercise, left- and right-multiplying by appropriate matrices having only one nonzero entry, we see that contains matrices having a nonzero entry in and 0 elsewhere for any choice of . Finally, multiplying by an appropriate scalar matrix , in fact contains every matrix of the form having a 1 in entry and 0 elsewhere for all . In particular, contains the ideal generated by such matrices.
So the kernel of is precisely the ideal generated by the matrices . In fact, since each is row- and column- equivalent to , the kernel of is the principal ideal generated by the matrix with a 1 in entry and 0 elsewhere.
Recall that every square matrix over is row- and column-equivalent to a diagonal matrix, and that two matrices and are row- and column-equivalent precisely when we have for some invertible matrices and . In particular, if and only if a diagonal matrix to which is row- and column equivalent is in . Moreover, we can assume that these diagonal matrices have only 1 or 0 on the main diagonal.
We claim that a diagonal matrix which has more than one nonzero entry on the main diagonal is not in . To see this, recall that . If is a diagonal matrix with more than one nonzero entry on the main diagonal, then certainly . As we saw in this previous exercise, every element of and has at most one nonzero entry on the main diagonal. So it remains to be seen that . Suppose and and consider . If this matrix is diagonal with nonzero diagonal entry , then for all , so that we have at most one nonzero diagonal entry.
So is in the kernel of if and only if it is row- and column- equivalent to a diagonal matrix with only one nonzero entry.