Determinants · Question 2

Let S= \A= pmatrix 0 & 1 & c \ 1 & a & d \ 1 & b & e pmatrix : a,b,c,d,e \0,1\ and |A| \-1,1\ \, where |A| denotes the determinant of A. Then the number of elements in S is .

Answer

16

Solution

Expand along R1: |A|=-1(e-d)+c(b-a)=(d-e)+c(b-a)

Case c=0: d-e= 1 gives 2 choices of (d,e); a,b free (4) 8

Case c=1: (d-e)+(b-a)= 1. Each difference takes -1,0,1 in 1,2,1 ways; the sum is 1 in 4+4=8 ways

Total =8+8=16

Answer: 16