Question

Solve the following systems of homogeneous linear equations by matrix method:
$x+y-6z=0$
$x-y+2z=0$
$-3x+y+2z=0$

Answer 1

Given set of equations,

$x+y-6z=0$

$x-y+2z=0$

$-3x+y+2z=0$

Arranging the above equations in form of matrix and finding the coefficient of matrix, we get,

$A=\left[\begin{array}{ccc}1& 1& -6\\ 1& -1& 2\\ -3& 1& 2\end{array}\right]$

$R_2\to R_2-R_1$

$R_3\to R_3+3R_1$

$A\sim \left[\begin{array}{ccc}1& 1& -6\\ 0& -2& 8\\ 0& 4& -16\end{array}\right]$

$det\left(A\right)=0$ as 2nd and 3rd rows are identical.

as, $\rho \left(A\right)=2,$ submatrix is $\left[\begin{array}{cc}1& 1\\ 0& -2\end{array}\right]$

we get,

$x+y-6z=0$

$-2y+8z=0$

Let $z=k$

$y=4k$

$x=2k$