Matrices And Determinants Ques 2
- The system of linear equations
$ \begin{aligned} & x+y+z=2, \quad 2 x+3 y+2 z=5 \\ & 2 x+3 y+\left(a^2-1\right) z=a+1 \end{aligned} $
(2019 Main, 9 Jan I)
(a) has infinitely many solutions for $a=4$
(b) is inconsistent when $a=4$
(c) has a unique solution for $|a|=\sqrt{3}$
(d) is inconsistent when $|a|=\sqrt{3}$
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Answer:
Correct Answer: 2.(d)
Solution: (d) According to Cramer’s rule, here
$D=\left|\begin{array}{ccc} 1 & 1 & 1 \\ 2 & 3 & 2 \\ 2 & 3 & a^2-1 \end{array}\right|=\left|\begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 2 & 1 & a^2-3 \end{array}\right| $
( Applying $C_2 \rightarrow C_2-C_1$ and $C_3 \rightarrow C_3-C_1 $ )
$=a^2-3 \quad $ (Expanding along $ R_1$)
and
$D_1=\left|\begin{array}{ccc} 2 & 1 & 1 \\ 5 & 3 & 2 \\ a+1 & 3 & a^2-1 \end{array}\right|=\left|\begin{array}{ccc} 2 & 1 & 0 \\ 5 & 3 & -1 \\ a+1 & 3 & a^2-1-3 \end{array}\right| $
( Applying $C_3 \rightarrow C_3-C_2 $)
$ =\left|\begin{array}{ccc} 2 & 0 & 0 \\ 5 & 3-\frac{5}{2} & -1 \\ a+1 & 3-\frac{(a+1)}{2} & a^2-1-3 \end{array}\right| $
( Applying $C_2 \rightarrow C_2-\frac{1}{2} C_1 $)
$ =\left|\begin{array}{ccc} 2 & 0 & 0 \\ 5 & \frac{1}{2} & -1 \\ a+1 & \frac{5}{2}-\frac{a}{2} & a^2-4 \end{array}\right| $
$=2\left[\frac{1}{2}\left(a^2-4\right)+\left(\frac{5}{2}-\frac{a}{2}\right)\right] \quad\left[\text { Expanding along } R_1\right]$
$=2\left[\frac{a^2}{2}-2+\frac{5}{2}-\frac{a}{2}\right]=a^2-4+5-a=a^2-a+1$
Clearly, when $a=4$, then $D=13 \neq 0 \Rightarrow$ unique solution and
when $|a|=\sqrt{3}$, then $D=0$ and $D_1 \neq 0$.
$\therefore \quad $ When $|a|=\sqrt{3}$, then the system has no solution i.e. system is inconsistent.
BETA
