If a , b and c are all different from zero and | 1 + a 1 1 1 1 + b 1 1 1 1 + c | = 0 , then the value of 1 a + 1 b + 1 c is:

Question

If $a$ , $b$ and $c$ are all different from zero and $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0,$ then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is:

Smart Achievers · Accepted Answer

(3)

Ans. $- 1$

Explanation:

Given, $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0$

Expanding the determinant,

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

$(1 + a) (b + c + b c) - c - b = 0$

$b + c + b c + a b + a c + a b c - b - c = 0$

$a b + b c + c a = - a b c$

Dividing both sides by $a b c$ ,

$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = - 1$

Solution

Q.
If $a$ , $b$ and $c$ are all different from zero and $| \begin{matrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{matrix} |$ $= 0,$ then the value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ is:

1 $0$

2 $a b c$

3 $- 1$

4 $\frac{1}{a b c}$

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Solution

Q. If a, b and c are all different from zero and |1+a1111+b1111+c|=0, then the value of 1a+1b+1c is:

1 0

2 abc

3 -1

4 1abc