Introduction to Matrices

In 1985 Arthur Cayley presented the system of matrices called Theory of matrices. It was the latest way to solve the system of linear equation. It is undoubtedly the most powerful tool in mathematics. It is being used in various science as well as a business field like budgeting, cost estimation, sales projection, etc. Matrices are also used in cryptography.

Definition Matrix:

A matrix is a rectangular array of numbers or function. The numbers inside the matrix are called the elements or entries.

It is a systematic arrangement of the element and may represent vector or scalar quantity. Matrix is donated by capital letters and the element are enclosed in [] square brackets or () parenthesis. We can add, subtract, and multiply matrices together, under certain conditions.

The following are some examples of matrices:

A= $\begin {bmatrix} 1&3 \\ 2&4\end {bmatrix}$ B= $\begin {bmatrix} -2&6&5 \\ 8&9&3 \\ 1&7&-4\end {bmatrix}$

In the above example, the horizontal lines of the element are called rows of the matrix and the vertical lines of the element are called column of the matrix.

Thus, A has 2 rows and 2 columns while B has 3 rows and 3 columns.

Order of a Matrix

It means how many rows and columns are there in a matrix. In matrix, the total number of rows is donated by m and columns are donated by n i.e. (m*n)

For example: $\begin {bmatrix} a_{11}&a_{12}.....&a_{1n} \\ a_{21}&a_{22}.....&a_{2n} \\ a_{31}&a_{32}.....& a_{3n}\end {bmatrix}_{m*n}$

A11 means 1st row and 1st column. Similarly, A21 means 2nd row and 2nd column. The total number of elements i.e. rows and columns in a matrix is denoted by m*n.

Example 1: Consider the following example regarding the number of male and female workers in two shops I and II.

	Male Workers	Female Workers
1	10	18
2	25	28

Represent the above information in the form of a 2*2 matrix. What do the entry in the 2nd row and 2nd column represent?

Solution: The information in the form of 2*2 matrixes is follows:

$A=\begin {bmatrix} 10 & 18\\ 25 & 28\end {bmatrix}$

The entry in 2nd row and 2nd column represents female workers i.e. 28.

Example 2: If a matrix has 8 elements what are the possible order it can have?

Solution: As we know, the order of a matrix is m*n.

So, the element of the matrix is m*n.

So, the ordered pair of 8 natural number are : (1,8) (8,1) (4,2) (2,4).

Exercise

1. In the matrix $A=\begin {bmatrix}-7&6&5\\8&9&3\\1&-2&4 \end {bmatrix}$ writes:

1. The order of the element
2. The number of elements in a matrix
3. Write the element present at $A_{21}$ , $A_{22}$ , $A_{32}$ , $A_{23}$

2. If a matrix has 12 elements what are the possible order it can have?

3. If a matrix has 26 elements what are the possible order it can have? What if it has 7 elements?

Definition Matrix:

Order of a Matrix

Exercise

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