In the previous section, we completed a discussion on inverse trigonometric functions. In this chapter, we will see matrices.
Matrices are very useful tools in mathematics. They are used in other fields like science, engineering, management, sociology also. They help to solve many complicated problems in a simple and easy manner.
Some basics can be written in 8 steps:
1. First, we will see an example. It can be written in 2 steps:
(i) Consider three students in a class. Student A, student B and Student C.
•
Student A has five notebooks and two pens. Student B has seven notebooks and three pens. Student C has twelve books and two pens.
•
The above information can be written in the form of a matrix.
 |
| Fig.19.1 |
• Fig.19.1(a) above, shows the information as a matrix.
• Fig.b shows more details.
• Each student has his/her own unique row.
♦ First row is for student A.
♦ The second row is for student B.
♦ The third row is for student C.
• Each article has it’s own unique column.
♦ The first column is for notebooks.
♦ The second column is for pens.
• The matrix in the above fig.19.1, is named as X. We use capital letters for naming matrices.
(ii) The same information can be written as another matrix. This is shown in fig.19.2 below:
 |
| Fig.19.2 |
•
Here, each article has it’s own unique row.
♦ First row is for notebooks.
♦ Second row is for pens.
•
Each student has his/her own unique column.
♦ First column is for student A.
♦ The second column is for student B.
♦ The third column is for student C.
•
The matrix in the above fig.19.2, is named as Y.
2. Now we will see another example. It can be written in 2 steps.
(i) An industrialist has three factories. Factory I, Factory II and Factory III.
•
Factory I has 30 workers, Factory II has 18 workers and Factory III has 20 workers.
•
The above information can be written in the form of a matrix.
 |
| Fig.19.3 |
•
Fig.19.3(a) above, shows the information as a matrix.
•
Fig.b shows more details.
•
Each factory has it's own unique row.
♦ First row is for Factory I.
♦ The second row is for Factory II.
♦ The third row is for Factory III.
•
The "No. of workers" has it’s own unique column.
♦ The one and only one column is for the No. of workers.
•
The matrix in the above fig.19.3, is named as A. We use capital letters for naming matrices.
(ii) The same information can be written as another matrix. This is shown in fig.19.4 below:
 |
| Fig.19.4 |
• Here, the No. of workers has it’s own unique row.
♦ The one and only one row is for the No. of workers.
• Each factory has it's own unique column.
♦ First column is for Factory I.
♦ The second column is for Factory II.
♦ The third column is for Factory III.
• The matrix in the above fig.19.4, is named as B.
3. Definition 1:
We can define a matrix in 3 steps:
(i) A matrix is a rectangular/square array.
(ii) It is an array of numbers or functions.
(iii) The numbers/functions in the matrix can be called either elements or entries.
4. Some examples are shown in fig.19.5 below:
 |
| Fig.19.5 |
•
We see that:
♦ A has 2 rows and 3 columns.
♦ B has 3 rows and 3 columns.
♦ C has 3 rows and 2 columns.
5. Order of a matrix
This can be written in 3 steps:
(i) Suppose that, a matrix has m rows and n columns.
•
Then it is a matrix of order m × n.
•
In simple terms, we can say that, it is an m × n matrix (read as an m by n matrix)
(ii) So in the fig.19.5 above,
♦ A is a 2 × 3 matrix.
♦ B is a 3 × 3 matrix.
♦ C is a 3 × 2 matrix.
(iii) Note that, the number of elements in a matrix, is closely related to the order of that matrix.
♦ A is a 2 × 3 matrix. It has 6 elements.
♦ B is a 3 × 3 matrix. It has 9 elements.
♦ C is a 3 × 2 matrix. It has 6 elements.
6. General form of a matrix
This can be written in 5 steps:
(i) Consider a matrix of order m × n.
•
If m and/or n is large, there will be a large number of elements in that matrix.
•
Whatever be the number of elements, we do not have to worry. Because, each element of the matrix has a unique position. All we need, is to know two items:
♦ Row in which an element is situated.
♦ Column in which an element is situated.
(ii) We can use the letters i and j to indicate rows and columns.
♦ Rows are indicated by the letter i.
♦ Columns are indicated by the letter j.
(iii) Suppose that, for an element, value of i = 35 and that of j = 41.
•
Then it means:
The element is situated at the intersection of the 35th row and 41st column.
(iv) Consider the matrix A shown in fig.19.6 below:
 |
| Fig.19.6 |
In this fig.,
• The three vertical dots indicate that, some rows are omitted.
• The three horizontal dots indicate that, some columns are omitted.
• The three diagonal dots indicate that, some rows and columns are omitted.
• Such omissions will not affect the general form because, based on the pattern, we will be able to write any row or column.
• The reader is advised to become familiar with the pattern. He/she must be able to answer questions like:
♦ Why is an element marked as a23 ?
♦ Why is an element marked as ai2 ?
♦ Why is an element marked as amj ?
♦ Why is an element marked as ain ?
♦ Why is the last element marked as amn ?
(v) The general form of matrix, which is shown in fig.19.6, can be written in a compact form as: $A \,=\,[a_{ij}]_{m \times n}$.
• aij is the element which lies in the ith row and jth column.
• It is also called the (i,j)th element of A.
7. For our present discussion, we will be considering only those matrices in which,
♦ Elements are real numbers.
♦ Elements are functions which take real values.
8. Matrix form of Points in the Cartesian plane.
This can be written in steps:
(i) Consider the point (x,y) in the Cartesian plane. This point can be represented by a matrix.
• We can write that matrix, either as $\begin{bmatrix}
{x} \\
{y} \\
\end{bmatrix}
$ or as $\begin{bmatrix}
{x} &{y} \\
\end{bmatrix}
$.
• For example, P(2,5) can be represented either as $\begin{bmatrix}
{2} \\
{5} \\
\end{bmatrix}
$ or as $\begin{bmatrix}
{2} &{5} \\
\end{bmatrix}
$.
(ii) This gives us an easy method to represent the vertices of geometrical figures in the Cartesian plane.
• Fig.19.7 below shows a six sided polygon in the Cartesian plane.
 |
| Fig.19.7 |
• Fig.19.8 below shows the matrix form of the vertices. We can use either (a) or (b).
♦ In (a), each vertex has a unique column.
♦ In (b), each vertex has a unique row.
 |
| Fig.19.8 |
(iii) This method has immense applications in computer programs related to graphics.
We have seen the basic details about matrices. In the next section, we will see some solved examples.
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