`color{green} ✍️` There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.

(i) `color{green}{"The interchange of any two rows or two columns."`
`color{green} ✍️`Symbolically the interchange of `i^(th)` and `j^(th)` rows is denoted by `R_i ↔ R_j` and interchange of `i^(th)` and `j^(th)` column is denoted by `C_i ↔ C_j`.

For example, applying `R_1↔ R_2` to `A= [ (1,2,1),(-1,sqrt 3 , 1),(5,6,7)]` we get `[(-1, sqrt 3 , 1),(1,2,1),(5,6,7)]`

(ii) `color{green}{"The multiplication of the elements of any row or column by a non zero number."}`

`color{green} ✍️` Symbolically, the multiplication of each element of the `i^(th)` row by `k`, where `k ≠ 0` is denoted by `R_i → k R_i`.
The corresponding column operation is denoted by `C_i → kC_i`

For example, applying `C_3 -> 1/7 C_3 ` to `B= [ (1, 2,1),(-1, sqrt 3 , 1)]` we get `[(1, 2, 1/7),(-1, sqrt 3 , 1/7)]`

(iii) `color{green}{"The addition to the elements of any row or column,"}`

`color{green}{" the corresponding elements of any other row or column multiplied by any non zero number."}`

`color{green} ✍️` Symbolically, the addition to the elements of `i^(th)` row, the corresponding elements of `j^(th)` row multiplied by k is denoted by `R_i → R_i + kR_j`.

`color{green} ✍️` The corresponding column operation is denoted by `C_i → C_i + kC_j`.

For example, applying `R_2 → R_2 – 2R_1`, to `C= [(1,2),(2,-1)]` we get ` [(1,2),(0,-5)]`

`color{green} ✍️` If `A` is a square matrix of order `m`, and if there exists another square matrix `B` of the same order m,

such that `color{green}{AB = BA = I}`,

then `B` is called the inverse matrix of `A` and it is denoted by `A^(– 1)`. In that case A is said to be invertible.

For example, let `A= [(2,3),(1,2)]` and `B= [(2,-3),(-1,2)]` be two matrices.

`AB= [(2,3),(1,2)] [(2,-3),(-1,2)]`

`= [ (4-3, -6+6),(2-2 ,-3+4)]= [(1,0),(0,1)]=I`

`BA=[(1,0),(0,1)]=I` Thus `B` is the inverse of `A`, in other words `B = A^(– 1)` and `A` is inverse of `B`, i.e., `A = B^(–1)`


`color{red}{ul{"Note :"}}`

1. A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

2. If B is the inverse of A, then A is also the inverse of B.

`color{red}{" Inverse of a square matrix, if it exists, is unique"}`

`color{green} ✍️` Let `A = [a_(ij)]` be a square matrix of order `m,` If possible, let B and C be two inverses of `A.` We shall show that `B = C.`

Since B is the inverse of A

`AB = BA = I`

Since C is also the inverse of A

`AC = CA = I`

`color{green} ✍️` Thus `B = BI = B (AC) = (BA) C = IC = C`

`color{green} ✍️` If `A` and `B` are invertible matrices of the same order, then `(AB)^(–1) = B^(–1) A^(–1)`

`color{blue}{"Proof" :}`

`(AB) (AB)^(–1) =1`
`=>` `A^(–1) (AB) (AB)^(–1) =A^(–1)I` (Pre multiplying both sides by `A^(–1)`

`=>` `(A^(–1)A) B (AB)^(–1) =A^(–1)` (Since `A^(–1) I = A^(–1))`

`=>` ` IB (AB)^(–1) =A^(–1)`

`=>` `B (AB)^(–1) =A^(–1)`

`=>` `B^(–1) B (AB)^(–1) =B^(–1) A^(–1)`

`=>` `I (AB)^(–1) =B^(–1) A^(–1)`

Hence `color{blue}{(AB)^(–1) =B^(–1) A^(–1)}`

`color{green} ✍️` If `A` is a matrix such that `A^(–1)` exists.

● write `A = IA`

● Apply a sequence of row operation on `A = IA` till we get, `I = BA. `

The matrix B will be the inverse of A.

● Similarly, if we wish to find `A^(–1)` using column operations, then, write `A = AI` and apply a sequence of column operations on `A = AI` till we get, `I = AB`

`color{red}{"Remark :"}` In case, after applying one or more elementary row (column) operations on
`A = IA (A = AI),` if we obtain all zeros in one or more rows of the matrix A on L.H.S., then `A^(–1)` does not exist