Question

If a matrix has $24$ elements, what are the possible order it can have? What, if it has $13$ 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 $24$ elements, we have to find all the ordered pairs of natural numbers whose product is $24$.
The ordered pairs are$(1,24),(24,1),(2,12),(12,2),(3,8),(8,3),(4,6)$ and $(6,4)$
Hence, the possible orders of a matrix having $24$ elements are:
$1\times 24,24\times 1,2\times 12,12\times 2,3\times 8,8\times 3,4\times 6,6\times 4$
$(1,13)$ and $(13,1)$ are the ordered pairs of natural numbers whose product is $13$.
Hence, the possible orders of a matrix having $13$ elements are $1\times 13$ and $13\times 1$