Question

Using matrix method find the values of $\lambda$ and $\mu$ so that the system of equations
$2x-3y+5z=12$

$3x+y+\lambda z=\mu$
$x-7y+8z=17$
has (i) a unique solution; (ii) infinite solution and (iii) no solution.

Answer 1

Here

$△=11(\lambda -2)=0\Rightarrow \lambda =2$

${△}_1=11(3\lambda -\mu +1)=11(7-\mu )$

${△}_2=11(-2\lambda +\mu -3)=11(\mu -7)$

${△}_3=11(\mu -7)$

$△\ne 0$ i.e $\lambda \ne 2$ unique solutions

then infinite solutions as $△,{△}_1,{△}_2,{△}_2$ are all zero.

$\lambda =2$ but $\mu \ne 7$ then no solution.