Question

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

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

Answer 1

The correct option is A is inconsistent when $|a|=\sqrt{3}$

For the given system of equation, the augmented matrix can be written as
$[A:B]=\left[\begin{array}{cccc}1& 1& 1& 2\\ 2& 3& 2& 5\\ 2& 3& a^2-1& a+1\end{array}\right]$

Row operation by $R_2\to R_2-2R_1$ and $R_3\to R_3-2R_1$ we will get,

$[A:B]=\left[\begin{array}{cccc}1& 1& 1& 2\\ 0& 1& 0& 1\\ 0& 1& a^2-3& a-3\end{array}\right]$

Row operation by $R_3\to R_3-R_2$

$[A:B]=\left[\begin{array}{cccc}1& 1& 1& 2\\ 0& 1& 0& 1\\ 0& 0& a^2-3& a-4\end{array}\right]$

Now if $a=\pm \sqrt{3}$, then rank of matrix $A=2$ and rank of the augmented matrix $[A:B]=3$

$\therefore \text{rank of}A\ne \text{rank}[A:B]$ and the system of linear equation becomes incosistent.
Hence, for $|a|=\sqrt{3}$ the system of linear equation is inconsistent.

$Alternatesolution$ :

If we consider all the three equations representing a plane and if we substitute $a^2-1=2$, i.e., $a=\pm \sqrt{3}$ the last two planes become parallel in nature.
So the given system of linear equations does not have a solution for $a=\pm \sqrt{3}$ and will become inconsistent.