Question

Solve the following systems of homogeneous linear equations by matrix method:
$3x-y+2z=0$
$4x+3y+3z=0$
$5x+7y+4z=0$

Answer 1

Given set of equations,

$3x-y+2z=0$

$4x+3y+3z=0$

$5x+7y+4z=0$

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

$A=\left[\begin{array}{ccc}3& -1& 2\\ 4& 3& 3\\ 5& 7& 4\end{array}\right]$

$R_2\to 3R_2-4R_1$

$R_3\to 3R_3-5R_1$

$A\sim \left[\begin{array}{ccc}3& -1& 2\\ 0& 13& 1\\ 0& 26& 2\end{array}\right]$

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

as, $\rho \left(A\right)=2,$ submatrix is $\left[\begin{array}{cc}3& -1\\ 0& 13\end{array}\right]$

we get,

$3x-y+2z=0$

$13y+z=0$

Let $z=k$

$y=-\frac{k}{13}$

$x=-\frac{9k}{13}$