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The system of linear equation $x + y + z = 2, 2x + 3y + 2z = 5$, $2x + 3y + (a^2 -1)\,z = a + 1$ then
is inconsistent when $a = 4$
has a unique solution for $\left| a \right| = \sqrt 3 $
has infinitely many solutions for $a = 4$
inconsistent when $\left| a \right| = \sqrt 3 $
Solution
By applying Crammer's Rule
$D = \left| {\begin{array}{*{20}{c}}
1&1&1\\
2&3&2\\
2&3&{{a^2} – 1}
\end{array}} \right|$
$ = 3\left( {{a^2} – 1} \right) – 6 – 2\left( {{a^2} – 1} \right) + 4$
$ = {a^2} – 1 – 2 = {a^2} – 3$
If $\left| a \right| \ne \pm \sqrt 3 \Rightarrow $system has unique solution
If $\left| a \right| = \sqrt 3 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. \begin{array}{l}
x + y + z = 1\\
2x + 3y + 2z = 1\\
2x + 3y + 2z = \pm \sqrt 3 + 1
\end{array} \right\}$
Hence system is inconsistent for $\left| a \right| = \sqrt 3 $
