Question

$\left| \begin{matrix} 1 & x & y+z \\\ 1 & y & z+x \\\ 1 & z & x+y \\\ \end{matrix} \right|$ is equal to

• A. $0$
• B. $x$
• C. $y$
• D. $xyz$

Answer 1

Answer: (A)

$0$

Answer 2

Let $\Delta =\left| \begin{matrix} 1 & x & y+z \\\ 1 & y & z+x \\\ 1 & z & x+y \\\ \end{matrix} \right|$
Applying ${{C}_{3}}\to {{C}_{3}}+{{C}_{2}}$
$=\left| \begin{matrix} 1 & x & x+y+z \\\ 1 & y & x+y+z \\\ 1 & z & x+y+z \\\ \end{matrix} \right|$
$=(x+y+z)\left| \begin{matrix} 1 & x & 1 \\\ 1 & y & 1 \\\ 1 & z & 1 \\\ \end{matrix} \right|$
$=0$
$(\because \,\,\,{{C}_{1}}\,and\,\,{{C}_{3}}\,are\,identical)$