The matrix $\begin{bmatrix} 2 & 5 & -7 \\ 0 & 3 & 11 \\ 0 & 0 & 9 \end{bmatrix}$ is known as:

$G = \left\{ \begin{bmatrix} x & x \\ x & x \end{bmatrix} : x \in \mathbb{R} \setminus \{0\} \right\}$ is a group with respect to matrix multiplication. In this group,the inverse of $\begin{bmatrix} 1/3 & 1/3 \\ 1/3 & 1/3 \end{bmatrix}$ is

If $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$,$I$ is the identity matrix of order $2$,and $a, b$ are arbitrary constants,then $(aI + bA)^2$ is equal to

If $A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 2 \\ 2 & 3 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -5 & 7 & 1 \\ 1 & -5 & 7 \\ 7 & 1 & -5 \end{bmatrix}$,then $AB$ is equal to

Let $A$ and $B$ be two square matrices of order $3$ and $AB = O_{3}$, where $O_{3}$ denotes the null matrix of order $3$. Then,

Construct a $3 \times 2$ matrix whose elements are given by $a_{ij} = \frac{1}{2}|i - 3j|$.