GATE CS 2008 Q3 — Linear Algebra: Matrices, Determinants, System of Linear Equations

Let $n>1$. Consider an $n \times n$ matrix $M$ with its elements from $\mathbb{R}$. Let the vector $(0,1,0,0, \ldots, 0) \in \mathbb{R}^{n}$ be in the null space of $M$. Which of the following options is/are always correct?

Consider the system of linear equations given below. $$\begin{array}{c} a x+y=b \\ 16 x+a y=24 \end{array}$$ Suppose the values of $a$ and $b$ are chosen such that the system of linear equations produce multiple solutions. Then the product of $a$ and $b$ is $\_\_\_\_$. (answer in integer)

The determinant of a $4 \times 4$ matrix $A$ is $3$. The value of the determinant of $2 A$ is $\_\_\_\_$. (answer in integer)

Consider the given system of linear equations for variables $x$ and $y$, where $k$ is a real-valued constant. Which of the following option(s) is/are CORRECT? $$\begin{array}{l} x+k y=1 \\ k x+y=-1 \end{array}$$

​​​​If $A=\left(\begin{array}{cc}1 & 2 \\ 2 & -1\end{array}\right)$, then which ONE of the following is $A^{8}$ ?

Let $L$, $M$, and $N$ be non-singular matrices of order $3$ satisfying the equations $L^2 = L^{-1}$, $M=L^8$ and $N=L^2$. Which ONE of the following is the value of the deteminant of $(M-N)$?