# Matrices:

#### What is a Matrix:

A matrix is an arrangement of the numbers or functions in the form of a rectangular array. The numbers or functions of a matrix are generally called elements. The matrices are used for writing linear equations in a compact form and solve the equations easily.

Let a girl "A" has 5 pens, then we can write it in the form of a matrix as [5].

If the same girl "A" has 5 pens and 3 pencils, then we can write it in matrix form as [5 3].

Now let if girl "A" has 5 pens, 3 pencils and another girl "B" has 6 pens, 4 pencils, then we can write it in matrix form as $$\begin{bmatrix} 5 & 3 \\ 6 & 4 \\ \end{bmatrix}$$

Similarly, if girl "A" has 5 pens, 3 pencils, and 2 books. The girl "B" has 6 pens, 4 pencils, and 3 books. The girl "C" has 8 pens, 7 pencils, and 5 books, then we can write it in matrix form as $$\begin{bmatrix} 5 & 3 & 2 \\ 6 & 4 & 3 \\ 8 & 7 & 5 \\ \end{bmatrix}$$

#### Order of Matrix:

A matrix with "m" number of rows and "n" number of columns is called the order of $$(m \times n)$$ matrix.

#### Notation of Matrix:

$$A = \begin{bmatrix} 1 & 2 \\ \end{bmatrix}$$ It is a $$(1 \times 2)$$ matrix or it can be written as $$A_{1 \times 2}$$

$$B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix}$$ It is a $$(2 \times 2)$$ matrix or it can be written as $$B_{2 \times 2}$$

$$C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix}$$ It is a $$(2 \times 3)$$ matrix or it can be written as $$C_{2 \times 3}$$

$$D = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end{bmatrix}$$ It is a $$(3 \times 2)$$ matrix or it can be written as $$D_{3 \times 2}$$

$$E = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix}$$ It is a $$(3 \times 3)$$ matrix or it can be written as $$E_{3 \times 3}$$

#### Elements of a Matrix:

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1j} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2j} & \cdots & a_{2n} \\ a_{i1} & a_{i2} & \cdots & a_{ij} & \cdots & a_{in} \\ \vdots & \vdots& \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mj} & \cdots & a_{mn} \end{pmatrix}$$ This is a $$(m \times n)$$ matrix. Here $$a_{11}, a_{12}......a_{mn}$$ are called elements of matrix. Here $$a_{ij}$$ is a general element which is placed at $$i^{th}$$ row and $$j^{th}$$ column. In the lower suffix of any element, the first number represents the row, and the second number represents the column of the matrix.

Note: Any point $$(x, y)$$ can also be written in the form of a matrix as $$\begin{bmatrix} x & y \\ \end{bmatrix}$$ or $$\begin{bmatrix} x \\ y \\ \end{bmatrix}$$

For example, the point $$(3, 2)$$ can be written in the form of a matrix as $$\begin{bmatrix} 3 & 2 \\ \end{bmatrix}$$ or $$\begin{bmatrix} 3 \\ 2 \\ \end{bmatrix}$$

Example: Read the following information regarding the number of boys and girls students studying in three schools A, B, and C.

$$\begin{matrix} & Boys & Girls \\ School \ A & 30 & 35 \\ School \ B & 45 & 25 \\ School \ C & 25 & 28 \\ \end{matrix}$$ Write the above information in the form of a matrix, and find what does the element in the second row and the second column represent?

Solution: The matrix form of the given information. $$A = \begin{bmatrix} 30 & 35 \\ 45 & 25 \\ 25 & 28 \\ \end{bmatrix}$$ It is a $$(3 \times 2)$$ matrix, it can also be written as $$A_{3 \times 2}$$. The element in the second row and second column represents 25 girls students in school B.