Determinants can be floppy

(b) Now, let's compute the determinant of A 6 RM\" differently. Write A as 4111 A12 A = , (A21 A22) where A22 6 1R("_1)X(\"_1). Other blocks have appropriate dimensions. If all 75 0, then det(A) = 5:11 det (A22 - Anal111512) , (10) where A22 A21 til11 A12 is non-singular. Suppose you can swap two rows without any ops, but such a swap changes the sign of the determinant. (i) If computing det(A) using (10) takes 2" ops, nd a recurrence relation between 2n and zn_1. (ii) Find the dominant term in n in the expression of z\" as a function of 11. (iii) Will you use (10) to compute the determinant of a 100 x 100 matrix? Answer logically. (0) Suppose A 6 RM" is nonsingular. Then, the inverse of A can be computed elementwise as det (Bij) [Al] det(A) ' ij = (_1)t+3 where B25 is the matrix obtained by removing the j-th row and the i-th column of A. Count the number of ops (only dominant terms in n) required to compute the inverse of A using the above equation. For computing determinants, assume that you are using the scheme in part (b). Compare it with the number of ops you will take to compute an LU decomposition followed by forward and backward substitutions to compute A'l. Hint: To compute A'1 using LU decomposition and forward/backward substitutions, you calculate the columns x1, . . . ,xn of A'1 by solving Azq = ei, where e;- E R\" is a vector whose elements are all zeros except its i-th element that is unity