If a, b and c are all non-zero and |(1+a, 1, 1), (1, 1+b, 1), (1, 1, 1+c)| = 0 then prove that

Question

If a, b and c are all non-zero and \(\begin{vmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{vmatrix}\) = 0 then prove that \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +1 = 0\)

Answer 1

Expanding along Ry we get

a[b( 1 + c) + c] - 0 + 1 [be - 0] = 0

⇒ a(b + be + c) +1 (be) = 0

⇒ ab + abc + ca + be = 0

⇒ ab + bc + ca + abc = 0

Dividing both sides by abc, we get

\(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} +1 = 0\)

Hence Proved.