Question

Determine the value of $\lambda$ for which the following system of equations fail to have a unique solution.
$\lambda x+3y-z=1$
$x+2y+z=2$
$-\lambda x+y+2z=-1$
Does it have any solution for $\lambda =1$ ?

Answer 1

Writing them in matrix form

$\left|\begin{array}{ccc}\lambda & 3& -1\\ 1& 2& 1\\ -\lambda & 1& 2\end{array}\right|\left|\begin{array}{c}x\\ y\\ z\end{array}\right|=\left|\begin{array}{c}1\\ 2\\ -1\end{array}\right|$

for no solution $△$=0

$\left|\begin{array}{ccc}\lambda & 3& -1\\ 1& 2& 1\\ -\lambda & 1& 2\end{array}\right|$=0

$\lambda (4-1)-3(2+\lambda )-1(1+2\lambda )=03x-6-3\lambda -1-2\lambda =0\lambda =-7/2$

$\left|\begin{array}{ccc}1& 3& -1\\ 2& 2& 1\\ -1& 1& 1\end{array}\right|=1(2-1)-3(4+1)-1(2+2)=3-15-4=-16\ne 0$

$\left|\begin{array}{ccc}1& 3& -1\\ 1& 2& 1\\ -1& 1& 2\end{array}\right|=\left(1\right)(4-1)-3(2+1)-1(1+2)=3-9-3=-9\ne 0CofactorMatrixM=\left|\begin{array}{ccc}3& -3& 3\\ -7& +1& -4\\ 5& -2& -1\end{array}\right|M^T\left|\begin{array}{ccc}3& -7& 5\\ -3& 1& -2\\ 3& -4& -1\end{array}\right|\left|\begin{array}{c}x\\ y\\ z\end{array}\right|=\frac{1}{-9}\left|\begin{array}{ccc}3& -7& 5\\ -3& 1& -2\\ 3& -4& -1\end{array}\right|\left|\begin{array}{c}1\\ 2\\ -1\end{array}\right|=\frac{1}{-9}\left|\begin{array}{c}-16\\ 1\\ -4\end{array}\right|=\left|\begin{array}{c}+16/9\\ -1/9\\ +4/9\end{array}\right|x=\frac{16}{9},y=\frac{-1}{9},z=\frac{4}{9}$