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# Let L be an n n elementary unit lower triangular matrix, that is, L has 1's on the main diagonal, and zeros in all other positions except column k.

Let L be an n × n elementary unit lower triangular matrix, that is, L has 1’s on the maindiagonal, and zeros in all other positions except column k. The nonzero entries in column kappear in positions k to n (and the (k, k) entry is, as already mentioned, equal to 1). Assumethe following equation holds,P1*P2 · · · Pq*L = M*P1*P2 · · · Pq,where each Pi, i = 1, . . . q is a permutation matrix that encodes an exchange between rows liand mi, such that k + 1 <= li <= n and k + 1 <= mi <= n, and M is some other n × n matrix.Show that if the above equation holds, then M must also be an elementary lower triangularmatrix all of whose nonzero entries are either 1’s on the diagonal or lie in positions k to n ofcolumn k. Furthermore, show that the entries of column k of M must be precisely the entriesof column k of L after the q swaps given by Pq, . . . , P1 are applied to this column. In otherwords, show thatM(:, k) = P1 · · · PqL(:, k).Hint: Let Q = P1 *P2 · · · Pq. Write Q in block-matrix form, with two diagonal blocks of size k×kand (n − k) × (n − k) respectively and corresponding off-diagonal block. The first diagonalblock is known in closed form. Why? The two off-diagonal blocks are also known in closedform. Why? Use block matrix-multiplication to separately evaluate columns 1 : k and alsocolumns k + 1 : n of the left-hand side Q*L and the right-hand side M*Q and determine theform of M.