Consider the system of linear equations - x + y + 2z = 0 3x - ay +...

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

$⇒ {a}^{2} - 7a + 12 = 0 ⇒ a = 3, 4$

{D}_{x} = \left|\begin{cases}0 & 1 & 2 \\ 1 & - a & 5 \\ 7 & - 2 & - a\end{cases}\right| = 15a + 31

${D}_{x} \ne 0\text{for}a = 3, 4$

Similarly we can show ${D}_{y} \ne 0, {D}_{z} \ne 0\text{for}a = 3, 4$

$⇒ n\left({S}_{1}\right) = 2$

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

$⇒ n\left({S}_{2}\right) = 0$

Hence, $n\left({S}_{1}\right) = 2 \text{ and } n\left({S}_{2}\right) = 0$