Question

The system of linear equations.
$x + y + z = 2$
$2x + 3y + 2z = 5$
$2x + 3y + (a^2 - 1)z = a + 1$

• A. has infinitely many solutions for $a = 4$
• B. is inconsistent when $|a| = \sqrt{3}$
• C. is inconsistent when $a = 4$
• D. has a unique solution for $|a| = \sqrt{3}$

Answer 1

Answer: (B)

$D = \begin{vmatrix}1&1&1\\ 2&3&2\\ 2&3&a^{2}-1\end{vmatrix} =a^{2} -3 $
$ D_{1} = \begin{vmatrix}2&1&1\\ 5&3&2\\ a+1&3&a^{2}-1\end{vmatrix} =a^{2}-a+1 $
$ D_{2} = \begin{vmatrix}1&2&1\\ 2&5&2\\ 2&a+1&a^{2}+1\end{vmatrix} =a^{2}-3 $
$ D_{3} = \begin{vmatrix}1&1&2\\ 2&3&5\\ 2&3&a+1\end{vmatrix}=a-4 $
D = 0 at $| a | = \sqrt{3}$ but $D_3 = \pm \sqrt{3} - 4 \neq 0$
So the system is Inconsistant for $|a |= \sqrt{3}$