Let E be the elementary matrix obtained from I by some row operation p. Prove that for every matrix A of the same size as E the product A*E^T coincides with the matrix obtained from A by applying the operation p to the columns of A.

Show that for every symmetric matrix B there exists a sequence of elementary matrices E₁, E₂,...,E_n such that

E_m...E₂E₁BE₁^T E₂^T...E_m^T = S

where S is a diagonal matrix with 1's and 0's on the diagonal (that is each diagonal entry of A is 1 or 0, each entry outside the diagonal is 0). Notice that S=S². Deduce that

B=[(E_m...E₂E₁)^-1S]*[S{(E_m ...E₂E₁)^T}^-1]

Notice that the matrix in the first square backets is the transpose of the matrix in the second square brackets (use the theorem about transposes). This completes the proof.