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 1/a+1/b+1/c + 1 = 0. ​

Question

If a, b and c are all non-zero and \(\begin{vmatrix} 1+a& 1 & 1 \\[0.3em] 1 &1+b & 1 \\[0.3em] 1 & 1 & 1+c \end{vmatrix}\) = 0, then prove that \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\) = 0.

Answer 1

Let Δ = \(\begin{vmatrix} 1+a& 1 & 1 \\[0.3em] 1 &1+b & 1 \\[0.3em] 1 & 1 & 1+c \end{vmatrix}\)

Given that,

Δ = 0. 

We can write the determinant Δ as,

Recall that the value of a determinant remains same if we apply the operation Ri→ Ri + kR_j or C_i→ C_i + kC_j.

We have,

Δ = 0.