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)

$D=\left|\begin{array}{ccc}1& 1& 1\\ 2& 3& 2\\ 2& 3& a^2-1\end{array}\right|=a^2-3$
$D_1=\left|\begin{array}{ccc}2& 1& 1\\ 5& 3& 2\\ a+1& 3& a^2-1\end{array}\right|=a^2-a+1$
$D_2=\left|\begin{array}{ccc}1& 2& 1\\ 2& 5& 2\\ 2& a+1& a^2+1\end{array}\right|=a^2-3$
$D_3=\left|\begin{array}{ccc}1& 1& 2\\ 2& 3& 5\\ 2& 3& a+1\end{array}\right|=a-4$
D = 0 at $|a|=\sqrt{3}$ but $D_3=\pm \sqrt{3}-4\ne 0$
So the system is Inconsistant for $|a|=\sqrt{3}$