Question

If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals ______.

पर्याय

• a + b + c

• abc

• 1

• – 1

Answer 1

If a, b, c are non-zero real numbers and if the system of equations (a – 1)x = y + z, (b – 1)y = z + x, (c – 1)z = x + y, has a non-trivial solution, then ab + bc + ca equals abc.

Explanation:

Given system of equations can be written as

(a – 1)x – y – z = 0

– x + (b – 1)y – z = 0

– x – y + (c – 1)z = 0

For non-trivial solution, we have

`|(a - 1, -1, -1),(-1, b - 1, -1),(-1, -1, c - 1)|` = 0

R₂ `rightarrow` R₂ – R₃

`|(a - 1, -1, -1),(0, b, -c),(-1, -1, c - 1)|` = 0

C₂ `rightarrow` C₂ – C₃

`|(a - 1, 0, -1),(0, b + c, -c),(-1, -c, c - 1)|` = 0

Apply R₃ `rightarrow` R₃ – R₁

`|(a - 1, 0, -1),(0, b + c, -c),(-a, -c, c)|` = 0

`\implies` (a – 1)[bc + c² – c²]  – 1[a(b + c)] = 0

`\implies` (a  – 1)[bc] – ab – ac = 0

`\implies` abc – bc – ab – ac = 0

i.e ab + bc + ca = abc