First slide

System of linear equations

Question

Statement - I : The system of equations $x + y + z = 6, x - y + z = 2, 2 x - y + 3 z = 9$ has unique solution
statement -II : The system of equations $x + y + z = 1, 2 x + y + z = 2, x + 2 y + 2 z = 1$ has no solution

Moderate

A

only I is are true
B

only II is true
C

Both I, II are true
D

Neither I nor II are true

Solution

Statement : I
$[A B] = {[\begin{matrix} 1 & 1 & 1 & 6 \\ 1 & - 1 & 1 & 2 \\ 2 & - 1 & 3 & 9 \end{matrix}]}_{\begin{matrix} R_{2} - R_{1} \\ R_{3} - 2 R_{1} \end{matrix}} ~ [\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & - 2 & 0 & - 4 \\ 0 & - 3 & 1 & - 3 \end{matrix}] ~ {[\begin{matrix} 1 & 1 & 1 & 6 \\ 0 & - 2 & 0 & - 4 \\ 0 & 0 & 2 & 6 \end{matrix}]}_{2 R_{3} - 3 R_{2}}$
$\Rightarrow$ unique solutions
Statement : II Since, $[A B] = [\begin{matrix} 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 2 \\ 1 & 2 & 2 & 1 \end{matrix}]$
$\begin{matrix} R_{2} \to R_{2} - 2 R_{1} \\ R_{3} \to R_{3} - R_{1} \end{matrix}$ $~$ $[\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & - 1 & - 1 & 0 \\ 0 & 1 & 1 & 0 \end{matrix}]$
$R_{3} \to R_{3} + R_{2} ~ [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & - 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$
Rank $A B =$ Rank $A < 3,$ Infinitely solutions

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