First Principles AI

Solution

A homogeneous system has the form $AX=0$ .

\begin{cases} x+4y-2z=0\\ 2x+y+5z=0\\ 5x+2y+8z=0 \end{cases}

Row-reduction gives rank $3$ , so only the trivial solution exists:

\boxed{x=0,\; y=0,\; z=0}

\begin{cases} x_1+4x_2+2x_3=0\\ 2x_1+x_2-3x_3=0\\ 3x_1+2x_2-4x_3=0 \end{cases}

RREF gives:

\begin{aligned} x_1-2x_3&=0\\ x_2+x_3&=0 \end{aligned}

Let $x_3=t$ . Then $x_1=2t$ and $x_2=-t$ .

\boxed{(x_1,x_2,x_3)=t(2,-1,1)}

\begin{cases} x_1+2x_2-x_3=0\\ x_1-x_2+5x_3=0\\ 2x_1+x_2+4x_3=0 \end{cases}

RREF gives:

\begin{aligned} x_1+3x_3&=0\\ x_2-2x_3&=0 \end{aligned}

Let $x_3=t$ . Then $x_1=-3t$ and $x_2=2t$ .

\boxed{(x_1,x_2,x_3)=t(-3,2,1)}