Let S be the of all column matrices [ array ll b _ 1 b _ 2 b _ 3...

We find $D = 0$ & since no pair of planes are parallel, so there are an infinite number of solutions. So, the infinite solutions shall lie on a common line of intersection of these planes. Hence, we can write any plane as a linear combination of other two planes:

$\alpha{P}_{1} - \lambda{P}_{2} = {P}_{3}$
$⇒$ ${P}_{1} + 7{P}_{2} = 13{P}_{3}$ (by comparing coefficients of $x, y, z$)
$⇒$ ${b}_{1} + 7{b}_{2} = 13{b}_{3}$
(a) $D \ne 0 ⇒$ unique solution for any ${b}_{1}, {b}_{2}, {b}_{3}$
(b) $D = 0$ but ${P}_{1} + 7{P}_{2} \ne 13{P}_{3}$
(c) $D = 0$ Also as they are parallel planes, they will have at least one solution only if they are all coincident, so that ${b}_{2} = - 2{b}_{1}, {b}_{3} = - {b}_{1}$
So each of ${b}_{1}, {b}_{2}, {b}_{3}$ that satisfy ${b}_{1} + 7{b}_{2} = 13{b}_{3}$, they don't always need to satisfy ${b}_{2} = - 2{b}_{1}, {b}_{3} = - {b}_{1}$. Hence option is wrong.

(d) $D \ne 0 ⇒$ unique solution for any ${b}_{1}, {b}_{2}, {b}_{3}$