Question

If the number of elements in the matrix is 60 then how many different orders of matrices are possible?

Answer 1

A matrix is said to be in the order of $m \times n$ if it has m rows and n columns. Total number of elements is m + n. So, we check from 1 and check one by one if it divides N.
If it divides, it will be one possible order.
And also we can calculate the total possible way of order of matrices. By adding the factor of N.

Complete step by step solution:
Here in the given question, N = 60 and we know that order $= m \times n$ ways of order of matrix.
Factor of 60 = ${2^2}\times3\times5$
So we can add the above factors to get the value of the total possible ways.
And ${2^2} + 3 + 5 = 12$
So in this way it will have 12 possible ways
As 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 are factors of 60 therefore we can make sets like this :
1 and 60 multiplies to get 60 so 1st and 2nd sets are $(1 \times 60),(60 \times 1)$.
2 and 30 multiplies to get 60 so 3rd and 4th sets are $(2 \times 30),(30 \times 2)$.
3 and 20 multiplies to get 60 so 5th and 6th sets are $(3 \times 20),(20 \times 3)$.
4 and 15 multiplies to get 60 so 7th and 8th sets are $(4 \times 15),(15 \times 4)$.
5 and 12 multiplies to get 60 so 9th and 10th sets are $(5 \times 12),(12 \times 5)$.
6 and 10 multiplies to get 60 so 11th and 12th sets are $(6 \times 10),(10 \times 6)$.

Therefore if the number of elements in the matrix is 60 then the order or matrix will be 12.

Note: To check whether our ways of matrix is correct or not we can cross-check it by $m \times n = N$
Like, our I way is $(1 \times 60)$ and $1 \times 60 = 60$
Another example is $(5 \times 12)$ and $5 \times 12 = 60$
So both are correct.