Basics of Matrices

What are matrices ?

If Matrix $A={\left[a\right]}_{3\times 3}$ is defined by

$$\begin{gathered} a_{ij}=\left\{\begin{array}{l}2i+3j, i<j\\ 5, i=j \\ 3i-2j, i>j\end{array}\right. \end{gathered}$$, then number of elements in $A$ which are greater than $5$ is 

Find the number of $2\times 2$ matrices satisfying following conditions:

$\left(\mathrm{i}\right) a_{\mathrm{ij}}$ is $1$ or $-1$

$\left(\mathrm{ii}\right) a_{11}{a}_{21}+a_{22}{a}_{12}=0$

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?

Write the order of the given matrix $\left[\begin{array}{ccc}1& 2& 3\\ 4& 8& 10\end{array}\right].$

Write the order of the given matrix $\left[\begin{array}{ccc}2& 3& 7\end{array}\right].$

Write the order of the given matrix $\left[\begin{array}{c}3\\ 4\\ 7\end{array}\right].$

Write the order of the given matrix $\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right].$

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

What is a matrix?

Let $A=\left[a_{ij}\right]$ be a $2\times 2$ matrix with $a_{ij}=(-1{)}^j{i}^2.$ Then the matrix $A$ is given by

The matrix $\left[\begin{array}{cc}1& 0\\ 2& -1\\ 0& -2\end{array}\right]$ has the order

What are the possible orders of a matrix having $24$ elements.