Question

The number of solutions of equation
$x_2-x_3=1,-x_1+2x_3=2,x_1-2x_2=3$ is

• A. zero
• B. one
• C. two
• D. infinite

Answer 1

Answer: (A)

The system is $0x_1+x_2-x_3=1$
$-x_1+0x_2+2x_3=2$
$x_1-2x_2+0x_3=3$
$\Rightarrow \left[\begin{array}{ccc}0& 1& -1\\ -1& 0& 2\\ 1& -2& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ or $AX=B$
Clearly $|A|=0$
Now Adj $A=\left[\begin{array}{ccc}4& 2& 2\\ 2& 1& 1\\ 2& 1& 1\end{array}\right]$
$\therefore (\text{Adj}A)B\ne 0$
$\Rightarrow$ system is inconsistent