Question

The system $2\,x+3 y+z=5,3\,x+y+5\,z=7$ and $x+4\,y-2\,z=3$ has

• A. unique solution
• B. finite number of solution
• C. infinite solutions
• D. no solution

Answer 1

Answer: (D)

no solution

Answer 2

The given system can be written as $A X=B$, where
$A =\begin{bmatrix}2 & 3 & 1 \\\ 3 & 1 & 5 \\\ 1 & 4 & -2\end{bmatrix}, X=\begin{bmatrix}x \\\ y \\\ z\end{bmatrix}, B=\begin{bmatrix}5 \\\ 7 \\\ 3\end{bmatrix}$
$\therefore |A| =\begin{vmatrix}2 & 3 & 1 \\\ 3 & 1 & 5 \\\ 1 & 4 & -2\end{vmatrix}$
$=2(-2-20)-3(-6-5)+1(12-1)$
$=2(-22)-3(-11)+1(11)$
$=-44+33+11=0$
$\because |A| =0$
Hence, there exist no solution.