Question

7. This question involves the concept of determinants and partitioned matrices. Historically, determinants first arose in the context of solving systems of linear equations for one set of variables in terms of another. For example, if the coefficient matrix of the system u= ax + by v=cx + dy is invertible, then the equations can be solved for x and y in terms of u and v as au – cu 2= du - bv ad - bc y = ad - bc which we can write in determinant notation as u a u V d с V y a a b b c d] с In the late seventeenth and early eighteenth centuries these formulas were extended to higher-ordered systems by laboriously solving the systems directly and then searching for common patterns in the solutions. Once those patterns were discovered, they were used to define higher-order determinants in a way that would allow the solutions of the higher-order systems to be expressed as ratios of determinants. This motivated us to extend the concept of a determinant to higher-order matrices. (a) Suppose A is a square matrix of order n. An elementary product from A is defined to be a product of n entries from A, no two of which come from the same row or same column. Thus, if A = [aij], then each elementary product is expressible in the form a1j1 (2j2 ... Anjn (4) where the column indices form a permutation {31, 32, ..., jn} of the integers from 1 to n and the row indices are in natural order. i. How many elementary products are there? ii. Let P(11,12, ..., jn) denote the minimum number of interchanges required to put the permutation {j1, 12, ..., İn} in natural order. Explain how to assign a sign to each elementary product in (4). iii. Write a formula to compute the determinant of A.

Answer 1

permutation J1, J2, Jg -- Joy of the integers foom 1 to n and the (a) Aij, azja. A = [aij] Hence an -(0) elementary po lut produt can be written as the following, where the column indices from. anjin 3 с рр. {, , 12 - sow indices are in natural order nindices can be have my peremutation Hence there are mi elementary product so (1) P(iz...In) denotes the minimum of interchanges required to put the peremutation & nin natural order of Phis even then sign should be it ve & P is add them dign should be a ve The sign Convention will be, linh 1 ай), Чај,-ия уп (..)-in) (ii) det (A) = & Aco (-1) p C, , 12. -. 9n)? det (a) = co ai), 9222 d. Anja all permutation of {1, 2, - : ? 4043, iz--.9m? po an aiz eg 911922 - 922 a 22