Q.Write short notes on : (any two)

Short notes on all three topics (any two were required).

(a) Multiplication of two matrices is defined only when the first matrix's columns equal the second's rows; the $(i,j)$ entry is the row $i$ of the first times column $j$ of the second.

Concise notes on matrix multiplication, median and range (answer any two; all three given).

1. Multiplication of two matrices. Two matrices $A$ and $B$ can be multiplied only when they are conformable: the number of columns of $A$ must equal the number of rows of $B$. If $A$ is of order $m\times n$ and $B$ is $n\times p$, the product $AB$ has order $m\times p$. The element in the $i$-th row and $j$-th column of $AB$ is obtained by multiplying the $i$-th row of $A$ with the $j$-th column of $B$ and adding: $c_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}.$ Matrix multiplication is associative and distributive but not commutative in general ($AB\neq BA$).
2. Median. The median is the value of the middle item when the observations are arranged in ascending (or descending) order; it divides the distribution into two equal halves. For $n$ ungrouped values the median is the $\left(\dfrac{n+1}{2}\right)$th item when $n$ is odd, and the average of the $\dfrac{n}{2}$th and $\left(\dfrac{n}{2}+1\right)$th items when $n$ is even. It is a positional average, unaffected by extreme values.
3. Range. …