The matrix X for which $\begin{bmatrix}5 & 4\\1 & 1\end{bmatrix}X=\begin{bmatrix}1 & -2\\1 & 3\end{bmatrix}$ is

Let $A=[a_{ij}]$ be a matrix of order $3 \times 4$ such that $a_{ij}=\begin{cases}2i-3j, & i \ge j \\ 3i+j, & i<j\end{cases}$ then the sum of all the elements of the first two rows of this matrix is:

If $[2\ \ x\ \ 3]\begin{bmatrix}-1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1\end{bmatrix}\begin{bmatrix}x\\0\\-1\end{bmatrix}=[-4]$, then the value(/s) of $x$ is

Let $A = [a_{ij}]_{3\times3}$ be a matrix such that $a_{ij} = \dfrac{i-j}{i+j}$, then which of the following statements is/are TRUE?

A. A is a symmetric matrix

B. A is a skew symmetric matrix

C. $|A| = 0$

D. $|A| \neq 0$

Choose the correct answer from the options given below:

If A is a square matrix of order 3, B = kA and |B| = $x$|A| then,

If $A = \dfrac{1}{9}\begin{bmatrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{bmatrix}$, then

(Where $A^{-1}$ and $A^T$ represent inverse and transpose of matrix $A$ respectively)

Assume $P$, $Q$, $R$ and $S$ are matrices of order $2 \times m$, $k \times n$, $m \times 2$ and $2 \times 3$ respectively. The restrictions on $k$, $m$ and $n$, so that $PQ + RS$ is defined are