Question

Solve the following systems of homogeneous linear equations by matrix method:
$2x+3y-z=0$
$x-y-2z=0$
$3x+y+3z=0$

Answer 1

Given,

system of linear equations,

$2x+3y-z=0$

$x-y-2z=0$

$3x+y+3z=0$

Arranging the above equations in form of the matrix, we get,

$A=\left[\begin{array}{ccc}2& 3& -1\\ 1& -1& -2\\ 3& 1& 3\end{array}\right]$

determinant $|A|=\left|\begin{array}{ccc}2& 3& -1\\ 1& -1& -2\\ 3& 1& 3\end{array}\right|$

$=2(-1+2)-3(3+6)+1(1+3)$

$=2\left(1\right)-3\left(9\right)+1\left(4\right)$

$=-21\ne 0$

$\rho \left(A\right)=3$

Hence the system has a trivial solution

$x=y=z=0$ only