## Chapter:3

# What are Matrices ?

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.

- The horizontal lines of elements are said to be,
**rows**of the matrix. - The vertical lines of elements are said to be,
**columns**of the matrix.

## Order Of Matrix

A matrix having m rows and n columns is called a matrix of order **m × n.**

- Namely A = [a
_{ij}]m×n to indicate that A is a matrix of order**m × n.**

## Equality Of Matrix

Two matrices A = [a_{ij}] and B = [b_{ij}] are said to be equal if :

- They are of the same order.
- Each element of A is equal to the corresponding element of B , that is a
_{ij}= b_{ij}for all i and j .

## Types Of Matrices

**Row Matrix**

A matrix is said to be a row matrix if it has only one row.

**Square Matrix**

A matrix in which the number of rows are equal to number of columns.

**Diagonal Matrix**

A square matrix is said to be a diagonal matrix if all its non-diagonal elements are zero.

**Scalar Matrix**

A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal.

**Zero Matrix **

A matrix is said to be zero matrix or null matrix if all its elements are zero.

**Identity Matrix**

A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.

**Column Matrix**

A matrix is said to be column matrix if it has only one column.

## Transpose Of A Matrix

^{'}or A

^{T}.

## Symmetric And Skew Symmetric Matrices

A symmetric and skew symmetric matrix both are square matrices. The **symmetric matrix** is equal to its transpose. The **skew symmetric matrix is a matrix** whose transpose is equal to its negative.

## Operations On Matrices

There are certain operations on matrices, addition of matrices, multiplication of matrix by scalar and multiplication of matrices.

**Addition of matrices**

_{ij}] and B = [b

_{ij}] are two matrices of the same order, say m × n. Then, the sum of the two matrices A and B is defined as a matrix C = [c

_{ij}]m × n , where c

_{ij}= a

_{ij}+ bij for all possible values of i and j .

**Multiplication of matrices**

_{ij}] be an m × n matrix and B = [b

_{jk}] be an n × p matrix. Then the product of the matrices A and B is the matrix C of order m × p.

**Multiplication of a matrix by scalar **

_{ij}]m×n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.

## Properties

**Properties of matrix addition **

**Commutative Law:**If A = [a_{ij}] and B = [b_{ij}] are two matrices of the same order, say m × n, then**A + B = B + A.****Associative Law:**If A=[a_{ij}], B=[b_{ij}] and C=[c_{ij}] are three matrices of the same order, say m × n, then**(A + B) + C = A + (B + C).****Additive Identity:**Let A = [a_{ij}] be an m × n matrix and O be an m × n zero matrix, then**A + O = O + A = A.****Additive Inverse:**Let A = [a_{ij}]m×n be any matrix, then we have another matrix as -A = [-a_{ij}]m×n such that**A + (-A) = (-A) + A = O.**

**Properties of multiplication of matrices **

**Associative Law:**For any three matrices A,B and C. We have**(AB)C = A(BC).****The Distribution Law:**For any three matrices A,B and C.**A(B+C) = AB + AC and (A+B)C = AC + BC****Existence of Multiplicative Identity:**For every square matrix A, there exist an identity matrix of same order such that**IA = AI = A.**- Matrix multiplication is not commutative AB
**≠**BA

**Properties of scalar multiplication of a matrix**

- k(A+B) = kA + kB, where k is scalar.
- (k+l)A = kA + lA, where k and l are scalars.

## Invertible Matrices

- If A is a square matrix of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and denoted by A
^{-1} - Inverse of a square matrix, if it exists is unique.