# Determine the Order of Matrix

Before we determine the order of matrix, we should first understand what matrices are. Matrices are defined as a rectangular array of numbers or functions. Since it is rectangular array, it is 2-dimensional. Basically, a two dimensional matrix consist of number of rows (m) and number of columns (n).

\( A =\left[

\begin{matrix}

3 & 4 & 9\cr

12 & 11 & 35 \cr

\end{matrix}

\right] \)

\( B =\left[

\begin{matrix}

2 & -6 & 13\cr

32 & -7 & -23 \cr

-9 & 9 & 15\cr

8 & 25 & 7\cr

\end{matrix}

\right] \)

The two matrices shown above A and B. The general notation of a matrix is given as:

\( A = [a_{ij}]_{m × n} \), where \( 1 ≤ i ≤ m , 1 ≤ j ≤ n \) and \(i , j \in N \)

You can see that the matrix is denoted by an upper case letter and its elements are denoted by the same letter in the lower case. \( a_{ij} \) represents any element of matrix which is in \( i^{th}\) row and \( j^{th} \) column. Similarly,\( b_{ij} \) represents any element of matrix B.

So, in the matrices given above, the element \( a_{21} \) represents the element which is in the \( 2^{nd} \)row and the \( 1^{st} \) column of matrix A.

∴a_{21} = 12

Similarly, \( b_{32} = 9 , b_{13} = 13 \) and so on.

Can you write the notation of 15 for matrix B ?

Since it is in \( 3^{rd} \) row and \( 4^{th} \) column, it will be denoted by \( b_{34} \).

If the matrix has \( m \) rows and \( n \) columns, it is said to be a matrix of the order \(m × n\). We call this an m by n matrix. So, A is a 2 × 3 matrix and B is a 4 × 3 matrix. The more appropriate notation for A and B respectively will be:

\( A =\left[

\begin{matrix}

3 & 4 & 9\cr

12 & 11 & 35 \cr

\end{matrix}

\right]_{2 × 3}

\)

\( B =\left[

\begin{matrix}

2 & -6 & 13\cr

32 & -7 & -23 \cr

-9 & 9 & 15\cr

8 & 25 & 7\cr

\end{matrix}

\right]_{4 × 3}

\)

So, if you have to find the order of the matrix, count the number or its rows and columns and there you have it!

It is quite fascinating that the order of matrix shares a relation with the number of elements present in a matrix.

The order of a matrix is denoted by \(a \times b\), and the number of element in a matrix will be equal to the product of a and b.

In the above examples, A is of the order 2 × 3 .Therefore, the number of elements present in a matrix will also be 2 times 3, i.e. 6.

Similarly, the other matrix is of the order 4 × 3, thus the number of elements present will be 12 i.e. 4 times 3.

This gives us an important insight that if we know the order of matrix, we can easily determing the total number of elements, that the matrix has. The conclusion hence is:

If a matrix is of m × n order, it will have mn elements.

But is the converse of the previous statement true?

The converse says that: If the number of element is mn, so the order would be \(m \times n\). This is definitely **not true. **It is because the product of mn can be obtained by more than 1 ways, some of them are listed below:

\(mn × 1\)

\(1 × mn\)

\(m × n\)

\(n × m\)

For example- Consider the number of elements present in a matrix to be 12. Thus the order of a matrix can be either of the one listed below:

\(12 \times 1\), or \(1 \times 12\), or \(6 \times 2\), or \( 2 \times 6\), or \(4 \times 3\), or \(3 \times 4\).

Thus, we have 6 different ways to write the order of a matrix, for the given number of elements.

Let us now look at a way to create a matrix for a given function:

For \( P_{ij} = i-2j \) , let us construct a 3 × 2 matrix.

So, this matrix will have 6 elements as following:

\( P =\left[

\begin{matrix}

P_{11} & P_{12}\cr

P_{21} & P_{22} \cr

P_{31} & P_{32} \cr

\end{matrix}

\right] \)

Now, we will calculate the values of the elements one by one. To calculate the value of \( p_{11} \) , substitute \( i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j \) .

\( P_{11} = 1 – (2 × 1) = -1 \)

\( P_{12} = 1 – (2 × 2) = -3 \)

\( P_{21} = 2 – (2 × 1) = 0 \)

\( P_{22} = 2 – (2 × 2) = -2 \)

\( P_{31} = 3 – (2 × 1) = 1 \)

\( P_{32} = 3 – (2 × 2) = -1 \)

Hence,

\( P =\left[

\begin{matrix}

-1 & -3\cr

0 & -2 \cr

1 & -1 \cr

\end{matrix}

\right]_{3 × 2}

\)