Question

If x, y, z are all different from zero and `|(1 + x, 1, 1),(1, 1 + y, 1),(1, 1, 1 + z)|` = 0, then value of x^–1 + y^–1 + z^–1 is ______.

Options

• x y z

• x^–1 y^–1 z^–1 

• – x – y – z

• –1

Answer 1

If x, y, z are all different from zero and `|(1 + x, 1, 1),(1, 1 + y, 1),(1, 1, 1 + z)|` = 0, then value of x^–1 + y^–1 + z^–1 is –1.

Explanation:

We have, `|(1 + x, 1, 1),(1, 1 + y, 1),(1, 1, 1 + z)|` = 0

Applying C₁ → C₁ – C₃ and C₂ → C₂ – C₃

⇒ `|(x, 0, 1),(0, y, 1),(-z, -z, 1 + z)|` = 0

Expanding along R₁

x[y(1 + z) + z] – 0 + 1(yz) = 0

⇒ xy + xyz + xz + yz = 0

⇒ `1/x + 1/y + 1/z + 1` = 0   .....[Dividing by (xyz) on both sides]

⇒ `1/x + 1/y + 1/z` = –1