Question

Solve the following systems of homogeneous linear equations by matrix method:
$x+y-z=0$
$x-2y+z=0$
$3x+6y-5z=0$

Answer 1

Given set of equations,

$x+y-z=0$

$x-2y+z=0$

$3x+6y-5z=0$

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

$A=\left[\begin{array}{ccc}1& 1& -1\\ 1& -2& 1\\ 3& 6& -5\end{array}\right]$

$R_2\to R_2-R_1$

$R_3\to R_3-3R_1$

$A\sim \left[\begin{array}{ccc}1& 1& -1\\ 0& -3& 2\\ 0& 3& -2\end{array}\right]$

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

as, $\rho \left(A\right)=2,$ submatrix is $\left[\begin{array}{cc}1& 1\\ 0& -3\end{array}\right]$

we get,

$x+y-z=0$

$3y-2z=0$

Let $z=k$

$y=\frac{2k}{3}$

$x=\frac{k}{3}$

or, we have,

$x=k,y=2k,z=3k$