(b) $A = \left| {\,\begin{array}{*{20}{c}}{ - 1}&2&5\\2&{ - 4}&{a - 4}\\1&{ - 2}&{a + 1}\end{array}\,} \right|\, = \,\left| {\,\begin{array}{*{20}{c}}0&0&{a + 6}\\0&0&{ - a - 6}\\1&{ - 2}&{a + 1}\end{array}\,} \right|$
(Operating ${R_1} \to {R_1} + {R_3}$ and ${R_2} \to {R_2} - 2{R_3})$
= $\left| {\,\begin{array}{*{20}{c}}0&0&0\\0&0&{ - a - 6}\\1&{ - 2}&{a + 1}\end{array}\,} \right|$ (Operating ${R_1} \to {R_1} + {R_2}$)
When $a = - 6,\,A = \left| {\,\begin{array}{*{20}{c}}0&0&0\\0&0&0\\1&{ - 2}&{ - 5}\end{array}\,} \right|$ , $\therefore \,\,\rho (A) = 1$
Where $\rho (A) = $number of non-zero rows
When $a = 6,\,A = \left| {\,\begin{array}{*{20}{c}}0&0&0\\0&0&{ - 12}\\1&{ - 2}&7\end{array}\,} \right|$ ,$\therefore $ $\rho (A) = 2$
When $a = 1,\,A = \left| {\,\begin{array}{*{20}{c}}0&0&0\\0&0&{ - 7}\\1&{ - 2}&2\end{array}\,} \right|$, $\therefore $ $\rho (A) = 2$
When $a = 2,\,A = \left| {\,\begin{array}{*{20}{c}}0&0&0\\0&0&{ - 8}\\1&{ - 2}&3\end{array}\,} \right|$ , $\therefore $$\rho (A) = 2$.