Consider the system of linear equations - x + y + 2 z = 0 3 x - a y + 5 z = 1 2 x - 2 y - a z = 7 Let S 1 be…

For system to be inconsistent $D=\left|\begin{array}{ccc}-1& 1& 2\\ 3& -a& 5\\ 2& -2& -a\end{array}\right|=0$

$\Rightarrow a^2-7a+12=0\Rightarrow a=3, 4$

$D_x=\left|\begin{array}{ccc}0& 1& 2\\ 1& -a& 5\\ 7& -2& -a\end{array}\right|=15a+31$

${\mathrm{D}}_x\ne 0\text{ for }\mathrm{a}=3,4$

Similarly we can show ${\mathrm{D}}_y\ne 0, {\mathrm{D}}_z\ne 0\text{ for }\mathrm{a}=3,4$

 $\Rightarrow \mathrm{n}\left({\mathrm{S}}_1\right)=2$

Now for infinitely many solutions $D=0$also ${\mathrm{D}}_{\mathrm{x}}={\mathrm{D}}_{\mathrm{y}}={\mathrm{D}}_z=0$ which is not possible value of any real of $\mathrm{a}$, since $D=0 \& D_x=0$ have different solutions for $a$

$\Rightarrow \mathrm{n}\left({\mathrm{S}}_2\right)=0$

Hence, $\mathrm{n}\left({\mathrm{S}}_1\right)=2 \& \mathrm{n}\left({\mathrm{S}}_2\right)=0$