Question

If a matrix has $4$ elements, what are the possible orders it can have?

Answer 1

First we have to know, what is the order of a matrix. So, the order of the matrix is defined as the number of rows and columns. The entries or the elements are the numbers in the matrix and each number is known as an element. In simple words, by talking about finding the orders of the matrix having $4$ elements, we are to find, in how many ways we can make a matrix that contains only $4$ elements. So, we are to think and arrange the number of ways we can make such a matrix containing $4$ elements.

Complete answer:
So, we have to find the number of ways of making a matrix having different orders and number of elements as $4$.
We know that the order of a matrix is the product of the number of rows and number of columns of the matrix. So, if a matrix has m rows and n columns, the order is $m \times n$.
So, we first find the factors of the number $4$.
$4 = 2 \times 2$
So, expressing in exponential form, we have, $4 = {2^2}$.
So, there are three different methods to express the number $4$ as a product of its factors: $1 \times 4$, $2 \times 2$, and $4 \times 1$.
We can make a matrix that has only $1$ row and $4$ columns.
Then, the matrix will have $4$ elements.
Hence, the order of this matrix is $1 \times 4$.
Similarly, we can make a matrix that has $4$ rows and $1$ columns.
Then also, the matrix will have $4$ elements.
Hence, the order of this matrix is $4 \times 1$.
Then, we can make a matrix that has $2$ rows and $2$ columns.
Then, the matrix will have $4$ elements.
Hence, the order of this matrix is $2 \times 2$.
Therefore, we can make only these three matrices with order $1 \times 4,4 \times 1,2 \times 2$.

Note:
The concept of order of matrix is very considerate about matrix operations. We can add or subtract only matrices that have the same order. Matrices with orders like $n \times n$ are called square matrices (square matrix in singular). And, we can only multiply two matrices, if their orders are $l \times m$ and $m \times n$ respectively, that is the number of columns of the first matrix is equal to the number of rows of the other matrix. We must know the prime factorization method to find the factors of a number.