Matrix

Definition

A number arranged in rows and columns which enclosed in large brackets or round brackets with a rectangular arrangement is called matrix. It is represented by capital letters . Here, its plural form is matrices .

For eg ;

A = \(\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\) B = \(\begin{bmatrix} 2 & 1 & 3 \\ 6 & 4 & 2 \end{bmatrix}\)

C = \(\begin {bmatrix} 4 & 2 \\ -3 & 2 \\ 1 & 0\end{bmatrix}\)

Rows and colums of a matrix

Matrix is a rectangular arrangement of numbers which contain elements in the form of row and column .Here, a column is usually a vertical line and row is a horizontal line .

eg;

A = \(\begin{bmatrix} 3 & 2 \\ 1 & 5\end{bmatrix}\)

This matrix A has two rows and two columns .

Order of matrix

The number of rows and columns present in a matrix gives the order of a matrix .

eg;

A = \(\begin {bmatrix} 3 & 1 & 2 \\ 5 & 4 & 6 \end {bmatrix}\)

Matrix A has 2 rows and 3 columns So, its order is 2 x 3.And it i read as 2 by 3.

Types of Matrices.

1.Row Matrix

A matrix having only one row is known as a matrix row . It can have any number of columns .

eg; A = \(\begin {bmatrix} -3 & 1 & 2\end{bmatrix}\)

It has only one row .

B = \(\begin{bmatrix} -1 & 2\end {bmatrix}\)

It has only one row .

Column Matrix

2.A matrix having only one column but any number of rows is known as a column matrix .

eg; A =\(\begin{bmatrix} -1 \\2 \\ 3 \end{bmatrix}\) B = \(\begin {bmatrix} 1 \\ 2 \end{bmatrix}\)

Here, A and B both are matrics have only one column .

3.Rectangular Matrix

A matrix having unequal number of rows and columns is known as a rectangular matrix .But,the number of rows and columns should not be equal .

A = \(\begin{bmatrix} 2 & -3 & 4 \\ 7 & 0 & 1 \end{bmatrix}\)

Here matrix A has 2 rows and 3 columns is unequal .So, it is rectangular matrix .

B = \(\begin{bmatrix} 1 & -4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\)

4.Zero Matrix

A matrix having each of the elements zero ( 0 ) is known as a null or zero matrix .

O = \(\begin {bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\)

It is a null matrix which is of order 2 x 3.

Square Matrix

A matrix whose number of rows and columns are equal is called a square matrix .

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

Then, In matrix A and B the number of rows and columns both are equal . So, they are squar matrix.

5.Equal Matrices

A pair of matrices having same order and equal corresponding elements are said to be equal matrices .

eg

A = \(\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}\) B = \(\begin{bmatrix} 2 & 3 \\ 4 & 5\end{bmatrix}\)

Here, matrices A and B are said to be equal matrices as their order is same as well as their corresponding elements are equal .

6.Operation on matrices

In this level we shall only see how matrices are added and subtracted . Let's look at the following examples.

A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and B = \(\begin{bmatrix} 5 & 6 \\ 7 & 8\end{bmatrix}\)

A + B = ?

Here, as the order of both the matrices is 2 x 2. we can add them

then, A + B = \(\begin {bmatrix} 1 + 5 & 2 + 6 \\ 3 + 7 & 4 +8 \end{bmatrix}\) [ We add the corresponding elements ]

=\(\begin{bmatrix} 6 & 8\\ 10 & 12 \end{bmatrix}\)

Again,

A = \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) C = \(\begin{bmatrix} 5 \\ 6 \end{bmatrix}\)

A + C = ? [ Order is not equal so, no addition is possible ]

= \(\begin{bmatrix} 1 & 2 \\ 3 & 4\end{bmatrix}\) + \(\begin{bmatrix} 5 \\ 6 \end{bmatrix}\)

As the order of these two matrices A and C are not the same we can not add them .

Now,

B - A = ?

so, B - A

= \(\begin{bmatrix} 5 - 1 & 6 - 2 \\ 7 - 3 & 8 - 4 \end{bmatrix}\)

= \(\begin{bmatrix} 4 & 4 \\ 4 & 4\end{bmatrix}\)