Question

If a matrix has $18$ elements , what are the possible orders it can have? what, if it has $5$ elements?

Answer 1

We know that if a matrix is of the order $m\times n$ , it has $mn$ elements.
Thus, to find all the possible orders of a matrix having $18$ elements, we have to find all the ordered pairs of natural numbers whose product is $18$.
The ordered pairs are : $(1,18),(18,1),(2,9),(9,2),(3,6)\text{and}(6,3)$
Hence the possible orders of a matrix having $18$ elements are:
$1\times 18,18\times 1,2\times 9,9\times 2,3\times 6\text{and}6\times 3$
$(1,5)$ and $(5,1)$ are the only ordered pairs of natural numbers whose product is $5$.