Question

If $\left| \begin{matrix} y + z & x & y \\ z + x & z & x \\ x + y & y & z \end{matrix} \right|$ = k (x + y + z)(x–z)², then k is equal to –

• A. 2xyz
• B. 1
• C. xyz
• D. x²y²z²

Answer 1

Answer: (B)

1

Answer 2

R₁ ® R₁ + R₂ + R₃ ;

D = (x + y + z) $\left| \begin{matrix} 2 & 1 & 1 \\ z + x & z & x \\ x + y & y & z \end{matrix} \right|$ = (x + y + z) $\left| \begin{matrix} 1 & 1 & 1 \\ x & z & x \\ x & y & z \end{matrix} \right|$

= (x + y + z) [1(z²–xy)–1(xz – x²) + 1 (xy – xz)]

= (x + y + z) [z² – xy – xz + x² + xy – xz]

= (x + y + z) [z² + x² – 2xz]

= (x + y + z) (x – z)²

\ k = 1