Question

Let $S$ be the set of all column matrices $\begin{bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}$ such that $b_1 , b_2 , b_3 \ \in \mathbb R$ and the system of equations (in real variables)
$-x + 2y + 5z = b_1$
$2x - 4y + 3z = b_2$
$x - 2y + 2z = b_3$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\begin{bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix} \in S$ ?

• A. $x + 2y +3z = b_1 , 4y + 5z = b_2$ and $x + 2y + 6z = b_3$
• B. $x + y +3z = b_1 , 5x + 5y + 6z = b_2$ and $- 2 x - y - 3z = b_3$
• C. $- x + 2y -5z = b_1, 2x -4y +10z = b_2$ and $x - 2y + 5z = b_3$
• D. $x + 2y + 5z = b_1 , 2x + 3z = b_2$ and $x + 4y - 5 z = b_3$

Answer 1

Answer: (A), (D)

We find D = 0 & since no pair of planes are parallel, so there are infinite number of solutions.
Let $\alpha P_{1} + \lambda P_{2} = P_{3} $
$\Rightarrow P_{1} + 7P_{2} = 13P_{3}$
$ \Rightarrow b_{1} + 7 b_{2} = 13 b_{3} $
(A) $D \neq 0 \, \Rightarrow $ unique solution for any $b_1 , b_2 , b_3$
(B) D = 0 but $P_1 + 7P_2 \neq 13 P_3$
(C) D = 0 Also $b_2 = - 2b_1 , b_3 = - b_1$
Satisfies $b_1 + 7b_2 = 13 b_3$ (Actually all three planes are co-incident)
(D) $D \neq 0 $