Question

5. (2 pt) is the permutation 1 2 3 4 5 (3 2 1 5 4 ) odd or even? Explain. 6. (2 pt) Find the rank of the matrix (1011 10 211 A= 3 1 1 1 4 3 3 3

Answer 1

I An Even codd) number of Recall that wanspostions if and only if even codo) numa ponuntata A bram position ha cycle of length 2 . (For exampl abe (1,2,..n) if ola)=b and o(b)= a. Then ca.b) = (bia) is a lranspostion. 2) Given permutation (12345) 7 32 1 5 4 s And Now write permutations in Sremsposton forms 4 (113) (415) Here member of transpostions û z By recall geven permutation ñ even permutation 6 Aug Rank of a matinn i to reduce into a simple from generasy by Elementary yow operation 2) Row operation does not chenye the row reale. (column rank.) Now geven that allo117 o21 L4 3 3 3 now 1 0 1 - B-38, Appy NO 2 ) Ry - Ry-48 lol-2-2 1o3-1-1

Appy 23 ) 2Rz-h Ry-) 2 Ry-36 los II oo -5-5 oo-5-5 Ry - By-B lozi oo-5-5 Remx of A=3 = number of non ero rows R(A)=3 (Con Rank of Az number of lenearly Indepenant Element in sow space for (column space]