Let Be a triangular factorization of A; let matrices Q n , Q n-1 ,…….Q 0 be defined by Qn = 1 and And let L* k stand for the matrix Q k L k Q t k  Prove that: (i) Each Q k  is a permutation matrix...

Be a triangular factorization of A; let matrices Q_n, Q_n-1,…….Q₀
be defined by Qn = 1 and

And let L*_k
stand for the matrix Q_kL_kQ^t
_k
 Prove that:

(i) Each Q_k
 is a permutation matrix agreeing with I in the first k rows and columns.

(ii)

(iii) Each L^*
_k
is a lower triangular eta matrix whose eta column is the k the column.