`\color{green} ✍️` A matrix is an ordered rectangular array of numbers or functions.

`\color{green} ✍️` A matrix having `m` rows and `n` columns is called a matrix of order `m × n`

`\color{green} ✍️` A matrix having `m` rows and `n` columns is called a matrix of order `m × n`

If any matrix `[b_(ij)]_(mxxn)`

● Row matrix `color{red}{=> m =1}`

●Column matrix `color{red}{=> n=1}`

●Square matrix `color{red}{=> m=n}`

●Diagonal matrix `color{red}{=> b_(ij) = 0; \ \ i ne j}`

●Scalar matrix `color{red}{=> b_(ij) = 0; \ \ i ne j \ \ "and" b_(ij) = k; \ \ i=j } `

●Identity matrix `color{red}{ => b_(ij) = 0; \ \ i ne j \ \ "and" b_(ij) = 1; \ \ i=j } `

●Zero matrix `color{red}{=> b_(ij) = 0}`

● Row matrix `color{red}{=> m =1}`

●Column matrix `color{red}{=> n=1}`

●Square matrix `color{red}{=> m=n}`

●Diagonal matrix `color{red}{=> b_(ij) = 0; \ \ i ne j}`

●Scalar matrix `color{red}{=> b_(ij) = 0; \ \ i ne j \ \ "and" b_(ij) = k; \ \ i=j } `

●Identity matrix `color{red}{ => b_(ij) = 0; \ \ i ne j \ \ "and" b_(ij) = 1; \ \ i=j } `

●Zero matrix `color{red}{=> b_(ij) = 0}`

● Euality of matrix :

`color{blue}{A = B =>[a_(ij)] = [b_(ij)]}`

● Addition of matrix :

`color{blue}{A + B =>[a_(ij)] + [b_(ij)]}`

● Substraction of matrix :

`color{blue}{A - B =>[a_(ij)] - [b_(ij)]}`

`color{blue}{A = B =>[a_(ij)] = [b_(ij)]}`

● Addition of matrix :

`color{blue}{A + B =>[a_(ij)] + [b_(ij)]}`

● Substraction of matrix :

`color{blue}{A - B =>[a_(ij)] - [b_(ij)]}`

Multiplication by scalar :

`k [a_(ij)]_(mxxn) = [ka_(ij)]_(mxxn)`

Matrix Multiplication =>

(please see image)

`k [a_(ij)]_(mxxn) = [ka_(ij)]_(mxxn)`

Matrix Multiplication =>

(please see image)

`=>` Transpose `color{red}{A = [a_(ij)]_(m × n)}`, then `color{blue}{A′ = A^T= [a_(ji)]_(n × m)}`.

Properties of transpose :

(i) (A′)′ = A,

(ii) (kA)′ = kA′ (where k is any constant)

(iii) (A + B)′ = A′ + B′

(iv) (A B)′ = B′ A′

Properties of transpose :

(i) (A′)′ = A,

(ii) (kA)′ = kA′ (where k is any constant)

(iii) (A + B)′ = A′ + B′

(iv) (A B)′ = B′ A′

`color{blue}{"Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix `A = [a_(ij)]` is said to be symmetric

`color{red}{"If "A′ = A, \ \ "that is,"\ \ [a_(ij)] = [a_(ji)]}` for all possible values of i and j.

`color{blue}{"Skew Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix A = [aij] is said to be skew symmetric matrix

`color{red}{"if" A′ = – A,\ \ "that is" \ \ a_(ji) = – a_(ij)` for all possible values of `i` and `j.`

`\color{green} ✍️` A square matrix `A = [a_(ij)]` is said to be symmetric

`color{red}{"If "A′ = A, \ \ "that is,"\ \ [a_(ij)] = [a_(ji)]}` for all possible values of i and j.

`color{blue}{"Skew Symmetric Matrix :"}`

`\color{green} ✍️` A square matrix A = [aij] is said to be skew symmetric matrix

`color{red}{"if" A′ = – A,\ \ "that is" \ \ a_(ji) = – a_(ij)` for all possible values of `i` and `j.`

Example of Row transformation :