If $A$ is a square matrix of order $n$ and $A = kB$, where $k$ is a scalar, then $|A|=$

If matrix  $A\, = \,\left[ {\begin{array}{*{20}{c}}
  1&{3k + \frac{1}{3}} \\ 
  0&1 
\end{array}} \right]$, then the value of  $\mathop \Pi \limits_{k = 1}^{36} \,\left[ {\begin{array}{*{20}{c}}
  1&{3k + \frac{1}{3}} \\ 
  0&1 
\end{array}} \right]$ is equal to :-

Let $a, b$ and $c$ be three real numbers satisfying

$\left[\begin{array}{lll}a & b & c\end{array}\right]\left[\begin{array}{lll}1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7\end{array}\right]=\left[\begin{array}{lll}0 & 0 & 0\end{array}\right]$…………….$(E)$

$1.$ If the point $P(a, b, c)$, with reference to $( E )$, lies on the plane $2 x+y+z=1$, then the value of $7 a+b+c$ is

$(A)$ $0$ $(B)$ $12$ $(C)$ $7$ $(D)$ $6$

$2.$ Let $\omega$ be a solution of $x^3-1=0$ with $\operatorname{Im}(\omega)>0$. If $a=2$ with $b$ and $c$ satisfying $( E )$, then the value of $\frac{3}{\omega^a}+\frac{1}{\omega^b}+\frac{3}{\omega^c}$ is equal to

$(A)$ $-2$ $(B)$ $2$ $(C)$ $3$ $(D)$ $-3$

$3.$ Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^2+b x+c=0$, then

$\sum_{n=0}^{\infty}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)^n$ is

$(A)$ $6$ $(B)$ $7$ $(C)$ $\frac{6}{7}$ $(D)$ $\infty$

Give the answer question $1,2$ and $3.$

If $A = \left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right]$ and $I = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]$, then which one of the following holds for all $n \ge 1$, (by the principal of mathematical induction)

The matrix $\left[ {\begin{array}{*{20}{c}}2&\lambda &{ – 4}\\{ – 1}&3&4\\1&{ – 2}&{ – 3}\end{array}} \right]$is non singular, if