Question

Let $S$ be the set of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$ 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 $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\in S$ ?

• 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 $-2x-y-3z=b_3$
• 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 $x+4y-5z=b_3$

Answer 1

Answer: (A), (D)

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