Definition and properties of matrices with a single nonzero entry

Let S be a ring with identity 1 \neq 0. Let n be a positive integer and let A = [a_{i,j}] be an n \times n matrix over S. Let E_{i,j} be the element of M_n(S) whose (i,j) entry is 1 and whose other entries are all 0.

  1. Prove that E_{i,j}A is the matrix whose ith row is the jth row of A and all other rows are 0.
  2. Prove that AE_{i,j} is the matrix whose jth column is the ith column of A and all other columns are 0.
  3. Deduce that E_{p,q}AE_{r,s} is the matrix whose (p,s) entry is a_{q,r} and all other entries are 0.

  1. By definition, E_{i,j}A = [c_{p,q}], where c_{p,q} = \sum_{k=1}^n e_{p,k}a_{k,q}. Note that if p \neq i, then e_{p,k} = 0, so that c_{p,q} = 0. If p = i, then c_{p,q} = a_{j,q}; thus the ith row of E_{i,j}A is the jth row of A, and all other entries are 0.
  2. The proof for AE_{i,j} is very similar.
  3. By the above arguments, E_{p,q}A is the matrix whose pth row is the qth row of A, and all other entries are 0. Then E_{p,q}AE_{r,s} is the matrix whose sth column is the rth column of E_{p,q}A, which is all zeroes except for the pth row, whose entry is the (q,r) entry of A, and all other entries are zero.
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