Let $A$ be a $3\times 3$ matrix with $det\left(A\right)=4$. Let $R_i$ denote the $i^{th}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R_2\to 2R_2+5R_3$ on $2 A$, then $det\left(B\right)$ is equal to :

Consider the following system of equations: 

$x+2y-3z=a$

$2x+6y-11z=b$

$x-2y+7z=c$

where $a,b$ and $c$ are real constants. Then the system of equations :

Let $A$ be a $2\times 2$ real matrix with entries from $\left\{0, 1\right\}$ and $\left|A\right|\ne 0$. Consider the following two statements;

$\left(P\right)$ If $A\ne I_2$, then $\left|A\right|=-1$

$\left(Q\right)$ If $\left|A\right|=1$, then $tr\left(A\right)=2$

Where $I_2$ denotes $2\times 2$ identity matrix and $tr\left(A\right)$ denotes the sum of the diagonal entries of $A$. Then

Let $S$ be the set of all $\lambda \in R$ for which the system of linear equations

$2x-y+2z=2$

$x-2y+\lambda z=-4$

$x+\lambda y+z=4$

has no solution. Then the set $S$