Suppose A = PtLU, where P is a permutation matrix, L is a lower-triangular matrix with ones

Question:

Suppose A = PtLU, where P is a permutation matrix, L is a lower-triangular matrix with ones on the diagonal, and U is an upper-triangular matrix.
a. Count the number of operations needed to compute PtLU for a given matrix A.
b. Show that if P contains k row interchanges, then
det P = det Pt = (ˆ’1)k .
c. Use det A = det Pt det L det U = (ˆ’1)k det U to count the number of operations for determining
det A by factoring.
d. Compute det A and count the number of operations when