Matrices is the plural form of matrix, which is a rectangular array or table with rows and columns of numbers or components. They can have as many columns and rows as they like. Addition, scalar multiplication, multiplication, transposition, and other operations can be performed on matrices.

There are several restrictions to follow when executing these matrix operations, such as they can only be added or subtracted if they have the same number of rows and columns, and they can only be multiplied if the first and second columns and rows are identical.

Matrices are rectangular tables with varying numbers of rows and columns in which numbers, variables, symbols, or phrases are arranged. They are rectangular arrays with different operations like addition, multiplication, and transposition specified for them. The elements of the matrix are the numbers or entries in it. Matrix entries are divided into two types: horizontal rows and vertical columns.

**Definition**:

A matrix is a rectangular array of integers, variables, symbols, or expressions that are defined for subtraction, addition, and multiplication operations. The number of rows and columns of a matrix determines its size (also known as its order). The 6×4 represents the arrangement of a matrix having 6 rows and 4 columns and is read as 6 by 4. The provided matrix B, for example, is a 3×4 matrix and is expressed as [ B ]_{3×4}:

B= [ 2 -1 3 5

0 5 2 7

1 -1 -2 9 ]

**Notation of Matrices:**

There will be m×n elements in a matrix with m rows and n columns. The elements in the matrix are represented by the lower case letter and two subscripts representing the location of the element in the number of rows and columns in the same sequence, in this case,’aij‘, where i is the number of rows and j is the number of columns. The element in the third row and second column of the provided matrix A, for example, would be a 32, as seen in the matrix below:

A = [ a_{11} a_{12} a_{13}……a_{1n}

a_{21} a_{22} a_{23}…….a_{2n}

a_{31} a_{32} a_{33}…….a_{3n}

. . . .

. . . .

a_{m1} a_{m2} a_{m3}…… a_{m4} ]

**Calculate Matrix:**

By performing operations on matrices such as addition, subtraction, multiplication, and so on, we can solve them. The number of rows and columns determines how matrices are calculated. The number of rows and columns must be the same for addition and subtraction, while the number of columns in the first and second matrices must be equal for multiplication. The following are the basic operations that can be performed on matrices:

- Addition of Matrices
- Subtraction of Matrices
- Scalar Multiplication

- Multiplication of Matrices
- Transpose of Matrices

**Addition of Matrices:**

The addition of matrices is only possible if both matrices have the same number of rows and columns. We add the necessary elements while adding two matrices. (A + B) = [aij ] + [bij] = [aij + bij ], where j and j are the row and column numbers, respectively. For instance:

[ 2 -1 [ 0 2 = [ 0+2 -1+2 = [ 2 1

0 5] + 1 -2] 0+1 5+(-2) ] 1 3]

**Subtraction of Matrices:**

Matrices subtraction is also achievable if both matrices have the same amount of rows and columns. We subtract the corresponding elements when subtracting two matrices. Specifically, (A – B) = [aij] – [bij] = [aij – bij], where I and j are the row and column numbers, respectively. For instance:

[ 2 -1 – [ 0 2 = [ 2 -3

0 5 ] 1 -2 ] -1 7 ]

**Scalar Multiplication:**

Scalar multiplication involves obtaining the product of a matrix A with any number ‘c’ by multiplying each entry of the matrix A by c. Specifically, (cA) i j = c(A i j )

**Scalar Multiplication Properties in Matrices**

Matrix characteristics for scalar multiplication of any scalars K and l with matrices A and B are as follows:

K(A + B) = KA + KB

A (K + l) = KA + IA

-K(A) = -(KA) = K(-A)

1·A = A

(-1)A = -A

**Conclusion:**

A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an object in mathematics. Matrix is the plural form of matrix, which is a rectangular array or table with rows and columns of numbers or components. They can have as many columns and rows as they like.

The number of rows and columns of a matrix determines its size (also known as its order).

There will be m×n elements in a matrix with m rows and n columns. The elements in the matrix are represented by the lower case letter and two subscripts representing the location of the element in the number of rows and columns in the same sequence, in this case,’aij’, where i is the number of rows and j is the number of columns.

By performing operations on matrices such as addition, subtraction, multiplication, and so on, we can solve them. The following are the basic operations that can be performed on matrices:

Addition of Matrices, Subtraction of Matrices, Scalar Multiplication, Multiplication of Matrices, Transpose of Matrices.