Question

Find solution homogeneous system of linear equations if possible

3x1+x2+x3=0

2x1+2x2-4x3=0

2x1-3x2+5x3=0

Answer 1

$\Delta= \begin{vmatrix} 3 & 1 &1\\ 2 & 2&-4\\2&-2&5 \end{vmatrix}$ $=3(10-12)-1(10+8)+1(-6-4)=-34\ne0$

$\Delta _1=\begin{vmatrix} 0 & 1&1 \\ 0 & 2&-4\\0&-3&5 \end{vmatrix}=0$

$\Delta _2=\begin{vmatrix} 3 & 0 &1\\ 2 & 0&-4\\ 2&0&5 \end{vmatrix}=0$

$\Delta _3=\begin{vmatrix} 3 & 1&0\\ 2 & 2&0\\ 2&-3&0 \end{vmatrix}=0$

The solution is

$x_1=\frac{\Delta _1}{\Delta}=0$

$x_2=\frac{\Delta _2}{\Delta}=0$

$x_3=\frac{\Delta _3}{\Delta}=0$