If [D = | pmatrix 1 & 1 & 1 1 & 1 + x & 1 1 & 1 & 1 + y pmatrix |] for [x 0,y 0,] then D i…

Question

If $$D = \left| {\begin{pmatrix} 1 & 1 & 1  \
   1 & {1 + x} & 1  \
   1 & 1 & {1 + y}  \ \end{pmatrix} } \right|$$ for $$x \ne 0,y \ne 0,$$ then $D$ is :

Accepted Answer

Given, $$D = \left| {\begin{pmatrix} 1 & 1 & 1  \
   1 & {1 + x} & 1  \
   1 & 1 & {1 + y}  \

\end{pmatrix} } \right|$$

Apply $$\,\,\,{R^2} \to {R_2} - {R_1}$$ $$\,\,\,\,$$

and $$\,\,\,\,$$ $$R \to {R_3} - {R_1}$$

$$\therefore$$ $$\,\,\,\,\,D = \left| {\begin{pmatrix} 1 & 1 & 1  \
   0 & x & 0  \
   0 & 0 & y  \

\end{pmatrix} } \right| = xy$$

Hence, $D$ is divisible by both $x$ and $y$

Solutions · 1