Question

Solve the following system of equations

$X_1+X_2+X_3$ = 3

$X_1-X_2$ = 0

$X_1-X_2+X_3$ = 1

• A. Unique solution
• B. No solution
• C. Infinite number of solutions
• D. Only one solution

Answer 1

The correct option is A Unique solution

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

$R_2\to R_2-R_1,R_3\to R_3-R_1$

-$\left[\begin{array}{ccccc}1& 1& 1& :& 3\\ 0& -2& -1& :& -3\\ 0& -2& 0& :& -2\end{array}\right]$

$R_3\to R_3-R_2$

-$\left[\begin{array}{ccccc}1& 1& 1& :& 3\\ 0& -2& -1& :& -3\\ 0& 0& 1& :& 1\end{array}\right]$

rank(A) = rank (A : B) = 3

$\therefore$ unique solution

Alternative sol.

$\left|A\right|=\left|\begin{array}{ccc}1& 1& 1\\ 1& -1& 0\\ 1& -1& 1\end{array}\right|$

= $1(-1-0)-1(1-0)+1(-1+1)$

= -2 $\ne$ 0

$\Rightarrow$ Unique solution