Basics of Matrices

What are matrices ?

Construct a $2\times 2$ matrix $A={\left[a_{ij}\right]}_{2\times 2}$ where elements are given by $a_{ij}=\frac{1}{2}{\left(i-j\right)}^2$.

Construct a $2\times 2$ matrix $A={\left[a_{ij}\right]}_{2\times 2}$ where elements are given by $a_{ij}=\frac{1}{2}{\left(i+2j\right)}^2$.

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right] \text{and} B=\lambda A=\left[\begin{array}{cc}a& b\\ c& a+b+c\end{array}\right]$, then

Find the value of $a_{23}+a_{32}$ in the matrix $A={\left[a_{ij}\right]}_{3\times 3}$ where $a_{ij}=\left\{\begin{array}{l}\left|2i-j\right| ,i>j\\ -i+2j+3 , i\le j\end{array}\right.$

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

If a matrix has $8$ elements, what are the possible orders it can have?

Consider the following information regarding the number of men and women workers in three factories I, II and III

Represent the above information in the form of a $3\times 2$ matrix. What does the entry in the third row and second column represent?

Let matrix $A=\left[a_{ij}\right]$, what is $i$ and $j$ in matrix $A$?

If a matrix has $28$ elements, Then total number of possible orders it can have, is

If the matrix $A={\left[a_{ij}\right]}_{2\times 4}$, then find the number of elements of $A$.

The additive inverse of the matrix $A=\left[\begin{array}{ccc}1& 0& 5\\ 2& -3& 4\end{array}\right]$ exists.

Write a matrix $B=\left[b_{ij}\right]$ of order $2\times 3$ whose elements are $b_{ij}=\frac{{\left(i+2j\right)}^2}{2}$

Write a $3\times 3$ matrix $B=\left[b_{ij}\right]$ whose elements are $b_{ij}=\left(i\right)\left(j\right)$.

Find a matrix $A=\left[a_{ij}\right]$ of the order $2\times 3$ whose elements are $a_{ij}=\frac{1}{2}|2i-3j|$.

Find the matrix $A=\left[a_{ij}\right]$ of order $2\times 2$ whose element $a_{ij}=2i-3j$.

Find a matrix $A=\left[a_{ij}\right]$ of order $2\times 2$ whose element $a_{ij}=\frac{(i+2j{)}^2}{2i}$.

Find a matrix $A=\left[a_{ij}\right]$ of order $2\times 2$ whose element $a_{ij}=\frac{2i-j}{3i+j}$.

Find the possible orders of matrix with $6$ elements.

Order of matrix $\mathrm{E}=\left[\begin{array}{c}-2\\ 3\\ 0\end{array}\right]$ is $a\times b$. Find $a+b$.