Question

For the set of equations
$x_1+2x_2+x_3+4x_4=2$
$3x_1+6x_2+3x_3+12x_4=6$
The following statement is true:

• A. Only the trivial solutions $x_1=x_2=x_3=x_4=0$
• B. There are no solutions
• C. A unique non-trivial solution exists.
• D. Multiple non-trivial solutions exist.

Answer 1

The correct option is D Multiple non-trivial solutions exist.

$\left[\begin{array}{cccc}1& 2& 1& 4\\ 3& 6& 3& 12\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]\left[\begin{array}{c}2\\ 6\\ 0\\ 0\end{array}\right]$

$\left[A\right]\left[X\right]=\left[B\right]$

As rank of matrix A = rank of matrix (A/B) = 1 < number of variables (4), given set of equations have infinite number of solutions.