Any even integer
Any odd integer
Zero
Any integer
||a/(a+1) - b/(b+1) c/(c+1)|| + ||a+1b+1c+1 a b c+1( )^n+2a( )^n+1b( )^nc|| = 0
→||a/(a+1) - b/(b+1) c/(c+1)|| + ( )^n||a+1b+1c+1 a b c+1 a-bc|| = 0 ----(1)
let |A| = ||a/(a+1) - b/(b+1) c/(c+1)||
applying C2 → C2 + C3 = 2
||a/(a+1) - b/(b+1) c/(c+1)||
applying C1 → C1 - C2 and expanding gives → 2||0 a b 0 c/(c+1)||
∴ |A| = 4b(c+a)
let |B| = ||a+1b+1c+1 a b c+1 a-bc||
applying R1 ↔ R3 and R2 ↔ R3 gives → ||a-bc a+1b+1c+1||
clearly B = A^T → |B| = 4b(c+a) (∵|A^T| = |A|)
∴ from (1) 4b(c+a)[1 + ( )^n] = 0
but given that b(c+a) ≠ 0 → ( )^n = -1
∴ n is an odd integer